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Questions and Answers
What is the decimal equivalent of the hexadecimal number (2AE0B)16?
What is the decimal equivalent of the hexadecimal number (2AE0B)16?
Each hexadecimal digit corresponds to a block of 4 binary digits.
Each hexadecimal digit corresponds to a block of 4 binary digits.
True
What is the remainder when you divide 12345 by 8?
What is the remainder when you divide 12345 by 8?
1
In base conversion, the process terminates when the quotient is ______.
In base conversion, the process terminates when the quotient is ______.
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Match the numeric system with its base:
Match the numeric system with its base:
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Which of the following gives the octal expansion of (12345)10?
Which of the following gives the octal expansion of (12345)10?
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How do you convert the decimal number 10 to hexadecimal?
How do you convert the decimal number 10 to hexadecimal?
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The hexadecimal number (1E5)16 equals 485 in decimal.
The hexadecimal number (1E5)16 equals 485 in decimal.
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What is the binary equivalent of the decimal number 25?
What is the binary equivalent of the decimal number 25?
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The hexadecimal representation of the decimal number 23670 is 5C76.
The hexadecimal representation of the decimal number 23670 is 5C76.
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What is the decimal equivalent of the binary number (101011111)₂?
What is the decimal equivalent of the binary number (101011111)₂?
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The octal expansion (701)₈ is equal to ____ in decimal.
The octal expansion (701)₈ is equal to ____ in decimal.
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Match the following numerical systems with their base:
Match the following numerical systems with their base:
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Which of the following digits are used in the hexadecimal system?
Which of the following digits are used in the hexadecimal system?
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What is the decimal value of the octal number (111)₈?
What is the decimal value of the octal number (111)₈?
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The decimal equivalent of the binary number (11011)₂ is ____.
The decimal equivalent of the binary number (11011)₂ is ____.
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What is the result of 228 mod 119?
What is the result of 228 mod 119?
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21 ≡ -8 (mod 6) is true.
21 ≡ -8 (mod 6) is true.
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Convert the decimal number 25 into binary.
Convert the decimal number 25 into binary.
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In modular arithmetic, the expression 'a is congruent to b modulo m' can be written as a _____ (mod m) = b (mod m).
In modular arithmetic, the expression 'a is congruent to b modulo m' can be written as a _____ (mod m) = b (mod m).
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Match the following number systems with their respective bases:
Match the following number systems with their respective bases:
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A modular expression can result in a negative value.
A modular expression can result in a negative value.
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Identify if 53 ≡ 108 (mod 7). Is this statement true or false?
Identify if 53 ≡ 108 (mod 7). Is this statement true or false?
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What does the notation $3 | 12$ signify?
What does the notation $3 | 12$ signify?
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The expression $5
mid 12$ means that 5 divides 12 evenly.
The expression $5 mid 12$ means that 5 divides 12 evenly.
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If $a | b$ indicates that a divides b, what is the mathematical definition of this statement?
If $a | b$ indicates that a divides b, what is the mathematical definition of this statement?
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If $a | b$, then a is called a ______ of b.
If $a | b$, then a is called a ______ of b.
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Match the following terms with their definitions:
Match the following terms with their definitions:
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Which statement is true regarding the integers?
Which statement is true regarding the integers?
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If $7 | 42$, then 42 is a multiple of 7.
If $7 | 42$, then 42 is a multiple of 7.
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Provide an example of an integer a and b such that $a | b$ is true.
Provide an example of an integer a and b such that $a | b$ is true.
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What is the result of 9009 mod 223?
What is the result of 9009 mod 223?
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Which digits are used in the hexadecimal system?
Which digits are used in the hexadecimal system?
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The expression 7 ≡ 19 (mod 3) is true.
The expression 7 ≡ 19 (mod 3) is true.
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What does the notation 'a ≡ b (mod m)' indicate?
What does the notation 'a ≡ b (mod m)' indicate?
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The binary system uses only the digits 0 and 1.
The binary system uses only the digits 0 and 1.
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To convert an integer n to base b, the value of the rightmost digit is found by computing n mod _____ .
To convert an integer n to base b, the value of the rightmost digit is found by computing n mod _____ .
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Match the following values of 'a' with their corresponding outputs for 'a mod 6':
Match the following values of 'a' with their corresponding outputs for 'a mod 6':
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The decimal expansion of the integer that has (101010)₂ is ______.
The decimal expansion of the integer that has (101010)₂ is ______.
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Match the following numeric systems to their bases:
Match the following numeric systems to their bases:
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Which of the following is true: 21 ≡ −8 (mod 6)?
Which of the following is true: 21 ≡ −8 (mod 6)?
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What is the decimal equivalent of (11011)₂?
What is the decimal equivalent of (11011)₂?
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Find 'a div m' when a = -10101 and m = 333.
Find 'a div m' when a = -10101 and m = 333.
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The digits used in the octal number system are from _____ to _____ .
The digits used in the octal number system are from _____ to _____ .
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The octal system includes the digit 8.
The octal system includes the digit 8.
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The decimal equivalent of (5C76)₁₆ is ______.
The decimal equivalent of (5C76)₁₆ is ______.
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What is the result of calculating $3644 , mod , 645$ using modular exponentiation?
What is the result of calculating $3644 , mod , 645$ using modular exponentiation?
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In modular arithmetic, the expression 'a is congruent to b modulo m' can sometimes result in a negative value.
In modular arithmetic, the expression 'a is congruent to b modulo m' can sometimes result in a negative value.
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What does the variable 'q' represent in the context of division?
What does the variable 'q' represent in the context of division?
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To find $a , mod , d$, the remainder is calculated until what condition is reached?
To find $a , mod , d$, the remainder is calculated until what condition is reached?
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What is the result of the binary addition of (0100 0111)₂ and (0111 0111)₂?
What is the result of the binary addition of (0100 0111)₂ and (0111 0111)₂?
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The algorithm for division and modulus ensures that the divisor 'd' can be less than zero.
The algorithm for division and modulus ensures that the divisor 'd' can be less than zero.
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What is the final result of the product of $7644 , mod , 645$ in integer terms?
What is the final result of the product of $7644 , mod , 645$ in integer terms?
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How many possibilities are there if only one of the bride and groom is in the wedding picture?
How many possibilities are there if only one of the bride and groom is in the wedding picture?
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The total number of possible 7-place license plates with letters and numbers, allowing repetition, is 175,760,000.
The total number of possible 7-place license plates with letters and numbers, allowing repetition, is 175,760,000.
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What is the formula for the number of permutations of n things taken r at a time?
What is the formula for the number of permutations of n things taken r at a time?
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The number of ways to arrange 6 people around a circular table is given by ___!.
The number of ways to arrange 6 people around a circular table is given by ___!.
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Match the types of arrangements with their corresponding scenarios:
Match the types of arrangements with their corresponding scenarios:
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How many different ways can letters and numbers be arranged in a 7-place license plate if repetition is prohibited?
How many different ways can letters and numbers be arranged in a 7-place license plate if repetition is prohibited?
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For circular seating, seating arrangements are unique even after rotations.
For circular seating, seating arrangements are unique even after rotations.
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When placing only the bride in the wedding picture, the total number of placements is ___.
When placing only the bride in the wedding picture, the total number of placements is ___.
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How many ways can a committee of six be formed containing exactly three students from four available students and eight teachers?
How many ways can a committee of six be formed containing exactly three students from four available students and eight teachers?
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According to the Pigeonhole Principle, if you have more pigeons than holes, at least one hole will contain at least one pigeon.
According to the Pigeonhole Principle, if you have more pigeons than holes, at least one hole will contain at least one pigeon.
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What is the Generalized Pigeonhole Principle?
What is the Generalized Pigeonhole Principle?
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A tree diagram is an effective tool for displaying a sequence of events, especially when the number of choices is ______.
A tree diagram is an effective tool for displaying a sequence of events, especially when the number of choices is ______.
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Match the principles with their explanations:
Match the principles with their explanations:
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How many teachers must be chosen to form a committee with exactly three students?
How many teachers must be chosen to form a committee with exactly three students?
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In a scenario where we have 11 balls and 5 boxes, at least one box must contain at least 3 balls.
In a scenario where we have 11 balls and 5 boxes, at least one box must contain at least 3 balls.
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What is the total number of possible outcomes when flipping a coin twice?
What is the total number of possible outcomes when flipping a coin twice?
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What is the decimal expansion of the hexadecimal number (2AE0B)16?
What is the decimal expansion of the hexadecimal number (2AE0B)16?
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Each octal digit corresponds to a block of 4 binary digits.
Each octal digit corresponds to a block of 4 binary digits.
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What is the octal expansion of the decimal number 327?
What is the octal expansion of the decimal number 327?
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The hexadecimal digits extend from 0 to _____ and A to _____ in representation.
The hexadecimal digits extend from 0 to _____ and A to _____ in representation.
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Match the following hexadecimal digits with their decimal values:
Match the following hexadecimal digits with their decimal values:
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Which of the following represents the binary expansion of the decimal number (11 1110 1011 1100)₂?
Which of the following represents the binary expansion of the decimal number (11 1110 1011 1100)₂?
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In base conversion, the process continues until the quotient equals zero.
In base conversion, the process continues until the quotient equals zero.
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Convert the decimal number 255 to hexadecimal.
Convert the decimal number 255 to hexadecimal.
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What is the octal equivalent of the binary number (011 111 010 111 100)₂?
What is the octal equivalent of the binary number (011 111 010 111 100)₂?
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The hexadecimal number (3EBC)16 equals to the binary number (0011 1110 1011 1100)₂.
The hexadecimal number (3EBC)16 equals to the binary number (0011 1110 1011 1100)₂.
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What is the result of adding the binary numbers (1110)₂ and (1011)₂?
What is the result of adding the binary numbers (1110)₂ and (1011)₂?
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To convert from decimal to octal, group the binary digits into blocks of ____.
To convert from decimal to octal, group the binary digits into blocks of ____.
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Match the numerical base with the block size used during conversion:
Match the numerical base with the block size used during conversion:
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Which of the following binary numbers correctly converts to the decimal number 231?
Which of the following binary numbers correctly converts to the decimal number 231?
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The binary number (0001 0101 0101)₂ converts to 85 in decimal.
The binary number (0001 0101 0101)₂ converts to 85 in decimal.
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What is the binary representation of the octal number (1604)8?
What is the binary representation of the octal number (1604)8?
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How many possibilities are there when both the bride and groom are included in the wedding picture?
How many possibilities are there when both the bride and groom are included in the wedding picture?
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Only one of the bride or groom in the picture yields 40,320 possibilities.
Only one of the bride or groom in the picture yields 40,320 possibilities.
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What is the total number of possibilities if only one of the bride and groom is included in the wedding picture?
What is the total number of possibilities if only one of the bride and groom is included in the wedding picture?
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The process of calculating different seating arrangements for the wedding picture involves the ______ rule.
The process of calculating different seating arrangements for the wedding picture involves the ______ rule.
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Match the number of people chosen with their respective roles in the wedding picture calculations:
Match the number of people chosen with their respective roles in the wedding picture calculations:
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How is the total number of arrangements affected when considering one of the bride or groom?
How is the total number of arrangements affected when considering one of the bride or groom?
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In the total possibilities when both the bride and groom are included, there are 30 approaches overall.
In the total possibilities when both the bride and groom are included, there are 30 approaches overall.
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How many people can be selected for positions if one of the couple is excluded?
How many people can be selected for positions if one of the couple is excluded?
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How many ways are there to seat 6 people around a table if rotations are considered equivalent?
How many ways are there to seat 6 people around a table if rotations are considered equivalent?
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The Inclusion-Exclusion Principle can be used to count the number of ways to perform two procedures, accounting for overlapping methods.
The Inclusion-Exclusion Principle can be used to count the number of ways to perform two procedures, accounting for overlapping methods.
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What is the formula provided for calculating the total number of methods using the Inclusion-Exclusion Principle?
What is the formula provided for calculating the total number of methods using the Inclusion-Exclusion Principle?
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The total number of bit strings of length 8 that start with 1 is ______.
The total number of bit strings of length 8 that start with 1 is ______.
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Match the following counts of bit strings with their conditions:
Match the following counts of bit strings with their conditions:
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What is the total number of bit strings of length 8 that either start with 1 or end with 00?
What is the total number of bit strings of length 8 that either start with 1 or end with 00?
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The formula |A∪B| = |A| + |B| - |A∩B| is incorrect.
The formula |A∪B| = |A| + |B| - |A∩B| is incorrect.
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Calculate the final answer for the number of bit strings of length 8 starting with 1 or ending with 00.
Calculate the final answer for the number of bit strings of length 8 starting with 1 or ending with 00.
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How many ways can a committee of six be selected if it contains exactly three students from four available students and eight teachers?
How many ways can a committee of six be selected if it contains exactly three students from four available students and eight teachers?
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The Pigeonhole Principle guarantees that at least one hole will contain more than one pigeon when the number of pigeons is less than or equal to the number of holes.
The Pigeonhole Principle guarantees that at least one hole will contain more than one pigeon when the number of pigeons is less than or equal to the number of holes.
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What does the Generalized Pigeonhole Principle state?
What does the Generalized Pigeonhole Principle state?
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If there are 11 balls divided among 5 boxes, at least one box will contain at least ____ balls.
If there are 11 balls divided among 5 boxes, at least one box will contain at least ____ balls.
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Match the following outcomes of flipping a coin twice with their results:
Match the following outcomes of flipping a coin twice with their results:
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In how many ways can a committee of six be formed if it contains at least three students?
In how many ways can a committee of six be formed if it contains at least three students?
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A tree diagram is useful when the number of choices or events is large.
A tree diagram is useful when the number of choices or events is large.
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What is a disadvantage of using a tree diagram?
What is a disadvantage of using a tree diagram?
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How many different arrangements are possible for a band consisting of 4 boys if each can play all 4 instruments?
How many different arrangements are possible for a band consisting of 4 boys if each can play all 4 instruments?
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The first digit of a US telephone area code can be 1.
The first digit of a US telephone area code can be 1.
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How many outcomes are possible when a die is rolled four times?
How many outcomes are possible when a die is rolled four times?
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The number of distinct arrangements of the letters in the word 'Mississippi' is _____?
The number of distinct arrangements of the letters in the word 'Mississippi' is _____?
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Match the numbers of area codes with their conditions:
Match the numbers of area codes with their conditions:
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How many different seating arrangements are possible for 8 people in a row without any restrictions?
How many different seating arrangements are possible for 8 people in a row without any restrictions?
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The total number of possible arrangements of the letters in the word 'propose' is _____?
The total number of possible arrangements of the letters in the word 'propose' is _____?
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There are only 6 possible values when rolling a die once.
There are only 6 possible values when rolling a die once.
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What is the formula to calculate the total number of passwords possible with a minimum of one digit?
What is the formula to calculate the total number of passwords possible with a minimum of one digit?
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How many possible passwords are there with 6 characters without any digits?
How many possible passwords are there with 6 characters without any digits?
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When using the product rule to arrange a wedding picture with 6 positions including the bride, how many total arrangements can be made?
When using the product rule to arrange a wedding picture with 6 positions including the bride, how many total arrangements can be made?
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What is the final calculated value for the total number of valid passwords including at least one digit?
What is the final calculated value for the total number of valid passwords including at least one digit?
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In the context of this content, what does the variable P6 represent?
In the context of this content, what does the variable P6 represent?
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How many ways can a label consisting of both a letter and a number between 1 and 50 be created?
How many ways can a label consisting of both a letter and a number between 1 and 50 be created?
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What is the total number of possible candidates for a faculty position if there are 20 candidates from Harvard and 50 from UCLA?
What is the total number of possible candidates for a faculty position if there are 20 candidates from Harvard and 50 from UCLA?
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If a restaurant offers 18 meat dishes, 10 fish dishes, and 5 vegetarian dishes, how many dinners are available to choose from?
If a restaurant offers 18 meat dishes, 10 fish dishes, and 5 vegetarian dishes, how many dinners are available to choose from?
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Using the product rule, how many different orders of ice cream can be made with 31 flavors, 4 sizes, and a choice of cone or dish?
Using the product rule, how many different orders of ice cream can be made with 31 flavors, 4 sizes, and a choice of cone or dish?
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Which rule applies when calculating the total number of ways to perform either procedure 1 or procedure 2?
Which rule applies when calculating the total number of ways to perform either procedure 1 or procedure 2?
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What is the value of $26 \times 50$?
What is the value of $26 \times 50$?
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In the context of choosing meals, if there are choices between types that are mutually exclusive, what does this signify?
In the context of choosing meals, if there are choices between types that are mutually exclusive, what does this signify?
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If a job candidate can only be from either Harvard or UCLA, which counting principle is being applied?
If a job candidate can only be from either Harvard or UCLA, which counting principle is being applied?
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What is the first step in converting a decimal integer n to a base b expansion?
What is the first step in converting a decimal integer n to a base b expansion?
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Which binary number corresponds to the octal digit 7?
Which binary number corresponds to the octal digit 7?
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If converting the binary number (11 1110 1011 1100)₂, which of the following represents the hexadecimal expansion?
If converting the binary number (11 1110 1011 1100)₂, which of the following represents the hexadecimal expansion?
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How many unique ways can 6 people be seated around a table considering rotations?
How many unique ways can 6 people be seated around a table considering rotations?
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What is the correct method to get the decimal expansion from the hexadecimal number (1A3)₁₆?
What is the correct method to get the decimal expansion from the hexadecimal number (1A3)₁₆?
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Using the Inclusion-Exclusion Principle, how many ways can procedures 1 and 2 be performed if there are 10 ways for procedure 1, 15 for procedure 2, and 5 equivalent ways?
Using the Inclusion-Exclusion Principle, how many ways can procedures 1 and 2 be performed if there are 10 ways for procedure 1, 15 for procedure 2, and 5 equivalent ways?
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Which statement is true regarding the conversion process between binary and octal?
Which statement is true regarding the conversion process between binary and octal?
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Which equation best illustrates the conversion from the decimal number 12345 to its octal equivalent?
Which equation best illustrates the conversion from the decimal number 12345 to its octal equivalent?
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During the conversion of (12345)₁₀ to octal, which of the following remainders denotes the last digit?
During the conversion of (12345)₁₀ to octal, which of the following remainders denotes the last digit?
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In the principle of Inclusion-Exclusion, what does the term 'm' represent?
In the principle of Inclusion-Exclusion, what does the term 'm' represent?
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Which base conversion process involves successively dividing by the base until reaching zero?
Which base conversion process involves successively dividing by the base until reaching zero?
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How many bit strings of length 8 start with 1?
How many bit strings of length 8 start with 1?
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If a seating arrangement can lead to 720 total arrangements before accounting for rotation, how many unique arrangements are there when factoring rotation?
If a seating arrangement can lead to 720 total arrangements before accounting for rotation, how many unique arrangements are there when factoring rotation?
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Using Inclusion-Exclusion, if a scenario has |A| = 128 and |B| = 64, what is |A ∩ B| if the total is 160?
Using Inclusion-Exclusion, if a scenario has |A| = 128 and |B| = 64, what is |A ∩ B| if the total is 160?
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How many outcomes are there when three coins are tossed?
How many outcomes are there when three coins are tossed?
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What is the total number of outcomes that contain at least two consecutive heads when tossing three coins?
What is the total number of outcomes that contain at least two consecutive heads when tossing three coins?
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When should the Inclusion-Exclusion Principle be applied in counting problems?
When should the Inclusion-Exclusion Principle be applied in counting problems?
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How many four-letter words can be formed from the letters G, R, O, U, P, S without repeating any letter?
How many four-letter words can be formed from the letters G, R, O, U, P, S without repeating any letter?
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How many ways can a committee of six be formed containing at least three students from four students and eight teachers?
How many ways can a committee of six be formed containing at least three students from four students and eight teachers?
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What is the number of outcomes if two six-faced distinct dice are thrown that yield a sum of either 3 or 6?
What is the number of outcomes if two six-faced distinct dice are thrown that yield a sum of either 3 or 6?
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What is the total number of outcomes when tossing three coins that do not include exactly two heads?
What is the total number of outcomes when tossing three coins that do not include exactly two heads?
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How many distinct outcomes are possible when throwing two six-faced dice?
How many distinct outcomes are possible when throwing two six-faced dice?
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In how many ways can a student answer questions on the test if they can leave answers blank?
In how many ways can a student answer questions on the test if they can leave answers blank?
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In how many ways can a committee of six be formed if it must contain exactly three students from four available students?
In how many ways can a committee of six be formed if it must contain exactly three students from four available students?
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What does the Pigeonhole Principle state about placing more than n pigeons in n holes?
What does the Pigeonhole Principle state about placing more than n pigeons in n holes?
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If there are 11 balls distributed among 5 boxes, what is the minimum number of balls that at least one box must contain?
If there are 11 balls distributed among 5 boxes, what is the minimum number of balls that at least one box must contain?
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What is a significant limitation of using tree diagrams in probability?
What is a significant limitation of using tree diagrams in probability?
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With the Pigeonhole Principle, what is true when dividing 13 balls into 4 boxes?
With the Pigeonhole Principle, what is true when dividing 13 balls into 4 boxes?
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Which of the following best describes a potential drawback of the tree diagram method?
Which of the following best describes a potential drawback of the tree diagram method?
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What is the average number of balls per box when placing 15 balls into 4 boxes?
What is the average number of balls per box when placing 15 balls into 4 boxes?
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If a committee of six includes at least three students, which scenario is NOT possible?
If a committee of six includes at least three students, which scenario is NOT possible?
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What is the octal representation of the binary number (011 111 010 111 100)₂?
What is the octal representation of the binary number (011 111 010 111 100)₂?
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Which of the following binary numbers corresponds to the hexadecimal number (3EBC)16?
Which of the following binary numbers corresponds to the hexadecimal number (3EBC)16?
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What is the binary equivalent of the octal number (1604)8?
What is the binary equivalent of the octal number (1604)8?
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What is the result of dividing $7644$ by $645$?
What is the result of dividing $7644$ by $645$?
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Which algorithm is used to add two binary numbers such as (1110)₂ and (1011)₂?
Which algorithm is used to add two binary numbers such as (1110)₂ and (1011)₂?
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What is the result of $3644 ext{ mod } 645$ using modular exponentiation?
What is the result of $3644 ext{ mod } 645$ using modular exponentiation?
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What is the binary equivalent of the hexadecimal number (ABBA)16?
What is the binary equivalent of the hexadecimal number (ABBA)16?
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In the context of the algorithm for division, what does 'r' represent?
In the context of the algorithm for division, what does 'r' represent?
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When applying the modular exponentiation, what is the first step after setting $x = 1$?
When applying the modular exponentiation, what is the first step after setting $x = 1$?
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Which of the following correctly converts the binary number (0110 1001 0001 0000)₂ to decimal?
Which of the following correctly converts the binary number (0110 1001 0001 0000)₂ to decimal?
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What binary representation results from summing (0100 0111)₂ and (0111 0111)₂?
What binary representation results from summing (0100 0111)₂ and (0111 0111)₂?
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What is the product of (1110 1111)₂ and (1011 1101)₂ expressed in binary?
What is the product of (1110 1111)₂ and (1011 1101)₂ expressed in binary?
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Which describes the process of converting the exponent in the modular exponentiation algorithm?
Which describes the process of converting the exponent in the modular exponentiation algorithm?
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What is the base of the number system used in the given binary numbers?
What is the base of the number system used in the given binary numbers?
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What does the notation $3 | 12$ represent?
What does the notation $3 | 12$ represent?
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If $5 ∤ 12$, what does this mean?
If $5 ∤ 12$, what does this mean?
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Which statement is true about the division operator?
Which statement is true about the division operator?
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Given integers $a$ and $b$, if $a | b$, which of the following can be concluded?
Given integers $a$ and $b$, if $a | b$, which of the following can be concluded?
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How can you express that an integer $a$ is not a factor of an integer $b$?
How can you express that an integer $a$ is not a factor of an integer $b$?
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Which of the following accurately describes divisibility?
Which of the following accurately describes divisibility?
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What is the total number of outcomes when two six-faced distinct dice are thrown if the sum of the digits shown is either 3 or 6?
What is the total number of outcomes when two six-faced distinct dice are thrown if the sum of the digits shown is either 3 or 6?
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If $a | b$ is true, which of the following cannot be concluded?
If $a | b$ is true, which of the following cannot be concluded?
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If a committee of six is to be formed from four students and eight teachers, how many different committees can be formed if at least three members must be students?
If a committee of six is to be formed from four students and eight teachers, how many different committees can be formed if at least three members must be students?
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Which of the following integers divides 12 evenly?
Which of the following integers divides 12 evenly?
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In tossing three coins, how many outcomes result in at least two consecutive heads?
In tossing three coins, how many outcomes result in at least two consecutive heads?
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What is the number of outcomes when three coins are tossed that do not contain at least two consecutive heads?
What is the number of outcomes when three coins are tossed that do not contain at least two consecutive heads?
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If a student can leave answers blank on a test with 10 questions, how many ways can a student answer the questions?
If a student can leave answers blank on a test with 10 questions, how many ways can a student answer the questions?
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In how many ways can a committee be formed that contains exactly three students from a group of four students and any combination of teachers?
In how many ways can a committee be formed that contains exactly three students from a group of four students and any combination of teachers?
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When tossing three coins, how many outcomes lead to exactly two heads?
When tossing three coins, how many outcomes lead to exactly two heads?
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How many ways can a committee of six be formed if it must include exactly three students from a group of four?
How many ways can a committee of six be formed if it must include exactly three students from a group of four?
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What does the Pigeonhole Principle state regarding the distribution of more than n items into n containers?
What does the Pigeonhole Principle state regarding the distribution of more than n items into n containers?
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When using a tree diagram, what is the main advantage mentioned about it?
When using a tree diagram, what is the main advantage mentioned about it?
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What is the generalized version of the Pigeonhole Principle regarding n items and k boxes?
What is the generalized version of the Pigeonhole Principle regarding n items and k boxes?
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If you have 11 items and are distributing them into 5 boxes, how many items must be in at least one of the boxes?
If you have 11 items and are distributing them into 5 boxes, how many items must be in at least one of the boxes?
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What is the disadvantage of using tree diagrams for outcomes?
What is the disadvantage of using tree diagrams for outcomes?
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In the context of forming a committee containing at least three students, what is the significance of the different student combinations?
In the context of forming a committee containing at least three students, what is the significance of the different student combinations?
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What is the outcome of removing 3 teachers from a committee selection of 8 if wanting at least three students?
What is the outcome of removing 3 teachers from a committee selection of 8 if wanting at least three students?
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How many distinct ways are there to seat 6 people around a round table?
How many distinct ways are there to seat 6 people around a round table?
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What is the principle of inclusion-exclusion primarily used for in combinatorics?
What is the principle of inclusion-exclusion primarily used for in combinatorics?
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If there are 128 bit strings of length 8 that start with 1, how many bit strings end with 00?
If there are 128 bit strings of length 8 that start with 1, how many bit strings end with 00?
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In the example of bit strings, how many start with 1 and end with 00 simultaneously?
In the example of bit strings, how many start with 1 and end with 00 simultaneously?
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Using the inclusion-exclusion formula, what is the total number of bit strings of length 8 that either start with 1 or end with 00?
Using the inclusion-exclusion formula, what is the total number of bit strings of length 8 that either start with 1 or end with 00?
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What does the formula |𝐴∪𝐵| = |𝐴| + |𝐵| − |𝐴 ∩ 𝐵| represent?
What does the formula |𝐴∪𝐵| = |𝐴| + |𝐵| − |𝐴 ∩ 𝐵| represent?
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What is the final answer calculated for the total number of bit strings that start with 1 or end with 00?
What is the final answer calculated for the total number of bit strings that start with 1 or end with 00?
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Considering the scenario of 8-bit strings, what is the maximum number of different arrangements possible without any restriction?
Considering the scenario of 8-bit strings, what is the maximum number of different arrangements possible without any restriction?
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Study Notes
EMath 105 Discrete Mathematics I for SE
- Course offered by the ECE Department
- Topics include Discrete Mathematics fundamentals and applications for SE students
Number Theory
- The integers and division are fundamental concepts in number theory
- Number theory involves specific notations, terminology, and theorems associated with integers and division, vital for various algorithms like hash functions, cryptography, digital signatures, and online security.
The Divides Operator
-
Notation: 3 | 12 (read as "3 divides 12")
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Specifies when an integer evenly divides another integer.
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Notation: 5 | -12 (read as "5 divides -12")
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Specifies when an integer evenly divides another integer.
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Notation: 5 ∤\nmid∤ 12 (read as "5 does not divide 12")
-
Specifies when an integer does not evenly divide another integer.
-
Key Definitions:
- a divides b (written as a | b): There is an integer c such that c * a = b
- Factor/Divisor of b: if a divides b, then a is a factor or divisor of b
- Multiple of a: if a divides b, then b is a multiple of a.
-
Examples:
- 3|-12 is True
- 3|7 is False
Results on the Divides Operator
- If a | b and a | c, then a | (b + c)
- If a | b, then a | bc for all integers c
- If a | b and b | c, then a | c
Divides Relation
- Va, b, c ∈\in∈ Z:
- a|0
- (a|b ∧\land∧ a|c) ⟹ \implies⟹ a | (b + c)
- a|b ⟹ \implies⟹ a|bc
- (a|b ∧\land∧ b|c) ⟹ \implies⟹ a|c
- Corollary: If a, b, c are integers, such that a | b and a | c, then a | mb + nc whenever m and n are integers.
Quotient and Remainder
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In division of a by d:
- q = quotient, r = remainder
- a = d × q + r
- 0 ≤ r < |d|
-
Example:
- 101 divided by 11:
101 = 11 × 9 + 2- q = 9
- r = 2
- 101 divided by 11:
-
More Examples:
- 25 mod 7 = 4
- 25 mod 5 = 0
- 35 mod 11 = 2
Modular Arithmetic
- If a and b are integers and m is a positive integer, then "a is congruent to b modulo m" if m divides a − b.
- Notation: a ≡ b (mod m).
- Rephrased: m | (a−b)
- Rephrased: a mod m = b mod m
Number Systems
- Decimal (base 10): digits 0-9
- Binary (base 2): digits 0-1
- Octal (base 8): digits 0-7
- Hexadecimal (base 16): digits 0-9, A-F (A=10, B=11, C=12, D=13, E=14, F=15)
Conversion to Base b
- Given an integer n and a base b>1, Algorithm used to find base b-representation
- Find n mod b to get the rightmost digit.
- Replace n with the quotient n/b
- Repeat
Binary, Octal, Hexadecimal Representation Conversion
- Conversion between binary, octal, and hexadecimal is straightforward using grouping.
- Octal digits correspond to three binary digits.
- Hexadecimal digits correspond to four binary digits
Algorithms for Integer Operations
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Addition Algorithm: Example for adding binary numbers provided (e.g., (1110)2 + (1011)2 = (11001)2
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Multiplication Algorithm: Example provided (e.g., (110)2 * (101)2 = (11110)2)
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Modular exponentiation: Algorithm to calculate bn mod m efficiently, important for large integers, Example given
-
Division and modulus algorithm
Combinatorial Analysis
-
Fundamental counting principles
- Product Rule: If there are n₁ ways to do the first step, and n₂ ways to do the second, then n₁n₂ are possible for both
- Sum Rule: If there are n₁ ways to do one task and n₂ ways to do a second task, the total ways to complete one task OR the other is n₁ + n₂
-
Counting Examples provided in the notes
-
Permutations: Formula
P(n,r) = n! / (n-r)! -
Circular arrangements: The number of circular arrangements of n distinct items is n-1
-
Inclusion-Exclusion Principle
-
Combinations: Formula
C(n,r) = n! / (r! (n-r)!)
Bit strings
- Problems involving bit strings with specific requirements (e.g., number of 1's) are outlined with examples
Exercises
- A collection of practice problems related to these concepts. Sample exercises are included in the notes (e.g., number of possible passwords, choosing a committee of students and faculty)
The Pigeonhole Principle
- Basic principle applied in counting problems, Example provided, and conditions
- If there are more pigeons than holes, at least one hole must have more than one pigeon
Trees Diagrams
-
Algorithm for listing all sequences of events, Example given for coin flip scenarios
-
Additional problems and exercises related to counting, permutations, and combinations in various scenarios are available in the notes
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Description
Test your knowledge on the conversion between different number bases including hexadecimal, decimal, octal, and binary. This quiz covers various concepts such as calculating remainders and matching systems with their bases. Challenge yourself with questions about number representation and conversions!
