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Questions and Answers
What is the total number of possible passwords of 6 to 8 characters that consist of upper case letters and digits, including at least one digit?
What is the total number of possible passwords of 6 to 8 characters that consist of upper case letters and digits, including at least one digit?
The number of possibilities for selecting a wedding picture with the bride not present is 90,720.
The number of possibilities for selecting a wedding picture with the bride not present is 90,720.
False
How many options are left for choosing the remaining five people after selecting the bride in the wedding picture scenario?
How many options are left for choosing the remaining five people after selecting the bride in the wedding picture scenario?
15120
The total number of passwords of length 7, which must contain at least one digit, is calculated as P7 = 36^7 - ______.
The total number of passwords of length 7, which must contain at least one digit, is calculated as P7 = 36^7 - ______.
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Match the following password lengths with their corresponding formulas:
Match the following password lengths with their corresponding formulas:
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How many ways can we label a chair if each label consists of both a letter and a number between 1 and 50?
How many ways can we label a chair if each label consists of both a letter and a number between 1 and 50?
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The sum rule can be used when the procedures are mutually exclusive.
The sum rule can be used when the procedures are mutually exclusive.
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If there are 31 flavors of ice cream, 4 sizes of serving, and a choice of cone or dish, how many different orders of ice cream can be placed?
If there are 31 flavors of ice cream, 4 sizes of serving, and a choice of cone or dish, how many different orders of ice cream can be placed?
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There are ___ candidates from UCLA and ___ candidates from Harvard.
There are ___ candidates from UCLA and ___ candidates from Harvard.
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Match the following examples with their corresponding counting rule:
Match the following examples with their corresponding counting rule:
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What is the total number of dinner options available at a restaurant with 18 meat dinners, 10 fish dinners, and 5 vegetarian dinners?
What is the total number of dinner options available at a restaurant with 18 meat dinners, 10 fish dinners, and 5 vegetarian dinners?
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If two procedures can occur simultaneously, the product rule should be used.
If two procedures can occur simultaneously, the product rule should be used.
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What does the product rule state in combinatorial counting?
What does the product rule state in combinatorial counting?
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How many teachers are available to form a committee?
How many teachers are available to form a committee?
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The Pigeonhole Principle states that if there are more holes than pigeons, at least one hole will contain more than one pigeon.
The Pigeonhole Principle states that if there are more holes than pigeons, at least one hole will contain more than one pigeon.
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What does the Generalized Pigeonhole Principle help to determine?
What does the Generalized Pigeonhole Principle help to determine?
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If we divide ___ balls among k boxes and n > k, at least one box contains at least ⌈n/k⌉ balls.
If we divide ___ balls among k boxes and n > k, at least one box contains at least ⌈n/k⌉ balls.
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Match the following concepts with their descriptions:
Match the following concepts with their descriptions:
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How many ways can a committee of exactly three students be formed from four students and eight teachers?
How many ways can a committee of exactly three students be formed from four students and eight teachers?
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In a Tree Diagram, the number of choices can never be large.
In a Tree Diagram, the number of choices can never be large.
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What could be a disadvantage of using Tree Diagrams?
What could be a disadvantage of using Tree Diagrams?
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How many distinct ways can 6 people be seated around a table considering rotations?
How many distinct ways can 6 people be seated around a table considering rotations?
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The Inclusion-Exclusion Principle states that to find the total number of ways to perform procedure 1 or procedure 2, you simply sum the ways without any adjustments.
The Inclusion-Exclusion Principle states that to find the total number of ways to perform procedure 1 or procedure 2, you simply sum the ways without any adjustments.
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What is the formula for the Inclusion-Exclusion Principle?
What is the formula for the Inclusion-Exclusion Principle?
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The total number of bit strings of length eight that start with 1 is _____.
The total number of bit strings of length eight that start with 1 is _____.
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Match the following terms with their corresponding numerical values:
Match the following terms with their corresponding numerical values:
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What is the value of |A ∪ B| for bit strings of length 8 that either start with 1 or end with 00?
What is the value of |A ∪ B| for bit strings of length 8 that either start with 1 or end with 00?
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The calculation for seating arrangements around a table considers both linear and circular permutations equally.
The calculation for seating arrangements around a table considers both linear and circular permutations equally.
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The number of bit strings ending with 00 is _____ for length eight.
The number of bit strings ending with 00 is _____ for length eight.
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How many total possibilities are there for a wedding picture that includes both the bride and groom?
How many total possibilities are there for a wedding picture that includes both the bride and groom?
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If only one of the bride and groom is included in the picture, what is the total number of possibilities?
If only one of the bride and groom is included in the picture, what is the total number of possibilities?
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The bride can occupy 5 positions in the wedding picture.
The bride can occupy 5 positions in the wedding picture.
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What is the product rule used for in this context?
What is the product rule used for in this context?
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If only the bride is included in the wedding picture, the total number of possibilities is _____.
If only the bride is included in the wedding picture, the total number of possibilities is _____.
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Match the following scenarios with their resulting possibilities:
Match the following scenarios with their resulting possibilities:
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How many people are there to choose from after placing the bride and groom when they are both included?
How many people are there to choose from after placing the bride and groom when they are both included?
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The sum rule is used when calculating possibilities for a scenario involving only one of the bride or groom.
The sum rule is used when calculating possibilities for a scenario involving only one of the bride or groom.
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How many different arrangements are possible for the band of four boys if each can play all four instruments?
How many different arrangements are possible for the band of four boys if each can play all four instruments?
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When rolling a die four times, the number of possible outcome sequences is 1296.
When rolling a die four times, the number of possible outcome sequences is 1296.
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How many area codes are possible in the US and Canada?
How many area codes are possible in the US and Canada?
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The number of ways to arrange the letters in the word 'arrange' is _____
The number of ways to arrange the letters in the word 'arrange' is _____
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Match each situation with the correct formula:
Match each situation with the correct formula:
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How many area codes can start with the digit 4?
How many area codes can start with the digit 4?
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The number of arrangements of the word 'Mississippi' is less than 2000.
The number of arrangements of the word 'Mississippi' is less than 2000.
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How many different letter arrangements can be made from the word 'propose'?
How many different letter arrangements can be made from the word 'propose'?
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Study Notes
Discrete Mathematics I for SE
- Course title: EMath 1105
- Course description: Discrete Mathematics I for students in the School of Engineering.
- Topics covered in the course:
- Combinatorial Analysis
- The Basics of Counting, Permutations, and Combinations
- Product rule:
- Procedure 1 AND procedure 2.
- n₁ ways to perform procedure 1
- n₂ ways to perform procedure 2
- Total ways: n₁ * n₂
- Sum rule:
- Procedure 1 OR procedure 2.
- n₁ ways to perform procedure 1
- n₂ ways to perform procedure 2
- Total ways: n₁+n₂
Counting Examples
-
Example 1: How many ways can we label a chair if each label consists of both a letter and a number between 1 and 50, inclusive? Given 26 letters and 50 numbers.
- Answer: 26 * 50 = 1300
-
Example 2: With 31 flavors of ice cream, 4 sizes of serving, and a choice of "cone" or "dish", how many different orders of ice cream are there?
- Answer: 31 * 4 * 2 = 248
Counting Examples: More
-
Example 1: One position for a faculty job at CPU. The applicant must be from either Harvard (20 candidates) or UCLA (50 candidates).
- Answer: Total candidates = 20 + 50 = 70
-
Example 2: A restaurant offers 18 dinners with meat, 10 with fish, and 5 vegetarian.
- Answer: Total choices = 18+10+5 = 33
Counting Examples: Passwords
-
Passwords consist of character strings of 6 to 8 characters. Each character is either an upper case letter or a digit. Each password must contain at least one digit. How many passwords are possible
- Find the total number of possible passwords by calculating the number of passwords with 6, 7 and 8 characters
- Calculate P6, P7, and P8 if there is no constraint in each case.
- Calculate the total number of possible outcomes without any digit restrictions.
- Calculate the total number of passwords.
- Calculate the difference between total # of possible options (without any digit constraint ) minus the total number of passwords without any digits.
-
Exclusions is #passwords without any digits
Counting Examples continued -
-
Example 1 (Rosen, section 4.1, question 38): There are 10 people (including the bride and groom). How many possibilities for a picture of 6 people if the bride must be in the picture?
- Calculate the total number of possible positions the bride can be
- Calculate the possibilities for the rest of the party via product rule considering available choices.
- Find total possibilities using product rule.
-
Example 2 (Rosen, section 4.1, question 38). How many possibilities if bride and groom must both be in the picture?
- Calculate the number of possibilities for bride and groom.
- Calculate the possibilities for the remaining members using product rule
- Find total possibilities using product rule.
-
Example 3 (Rosen, section 4.1, example 16): How many bit-strings of length 8 either begin with 1 or end with 00?
- Count strings starting with 1.
- Count strings ending with 00.
- Count strings starting with 1 and ending with 00
- Apply the inclusion-exclusion principle
- Find the total number of bit strings.
Exercises, and Answers
- Exercise and answers are also documented for exercises as outlined
Permutations
- Formula: P(n,r) = n! / (n-r)!
Combination
- Formula: C(n, r) = n! / (r! * (n-r)!)
Circular Seating
- Number of ways to seat n people around a circular table = (n-1)!
Principle of Inclusion-Exclusion (PIE)
- Formula: (n₁ + n₂ - m)
- n₁: ways for procedure 1
- n₂: ways for procedure 2
- m: ways of doing both procedure 1 and procedure 2 equally.
Tree Diagrams
- A systematic diagram showing all possible sequences of events in a branching format.
Probability
-
Example 1: How many possible outcomes can occur from rolling a six-sided die four times?
- Answer: 1296
- Possible outcomes by multiplication.
-
Example 2: How many possible arrangements are there of four instruments (John, Jim, Jay, Jack) if John and Jim can play all four, but Jay and Jack can only play piano and drums?
- Answer: 24
- Calculated using product rule and accounting for limited instruments per person
Additional problems:
- Telephone area codes in the U.S. (digits 2-9, 0/1, 1-9)
- Letter arrangements for words (fluke, propose, Mississippi, arrange)
- Seating arrangements for 8 people (with and without constraints)
- Handshakes between 20 people
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Description
Explore the foundational concepts of Discrete Mathematics as part of EMath 1105 for engineering students. This quiz covers topics such as combinatorial analysis, counting principles, permutations, and combinations through practical examples. Test your understanding of the product and sum rules in various counting scenarios.
