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How many different ways can 7 boys sit in a round table with the condition that two particular boys must sit together?
How many different ways can 7 boys sit in a round table with the condition that two particular boys must sit together?
What is the number of arrangements for forming a necklace from 50 different jewels?
What is the number of arrangements for forming a necklace from 50 different jewels?
If 3 ladies and 3 gents are seated at a round table where exactly two ladies must sit together, how many arrangements can be made?
If 3 ladies and 3 gents are seated at a round table where exactly two ladies must sit together, how many arrangements can be made?
How many unique arrangements can be made from the letters of the word 'DOGMATIC'?
How many unique arrangements can be made from the letters of the word 'DOGMATIC'?
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What is the number of arrangements of 10 different items taken 4 at a time, ensuring one specific item is always included?
What is the number of arrangements of 10 different items taken 4 at a time, ensuring one specific item is always included?
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How many permutations can be made from 10 different things taken 4 at a time where one specific item cannot be used?
How many permutations can be made from 10 different things taken 4 at a time where one specific item cannot be used?
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Mr. X and Mr. Y have six vacant seats in a railway compartment. How many ways can they occupy the seats?
Mr. X and Mr. Y have six vacant seats in a railway compartment. How many ways can they occupy the seats?
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How many numbers can be formed using the digits 1, 2, 3, 4, 5, 6, and 7 that lie between 100 and 1000?
How many numbers can be formed using the digits 1, 2, 3, 4, 5, 6, and 7 that lie between 100 and 1000?
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What is the formula to calculate the number of ways to arrange n persons at a round table?
What is the formula to calculate the number of ways to arrange n persons at a round table?
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If there are 6 different colors of beads, how many distinct necklaces can be formed using all the beads?
If there are 6 different colors of beads, how many distinct necklaces can be formed using all the beads?
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In circular permutations, how do we differentiate arrangements that are considered identical?
In circular permutations, how do we differentiate arrangements that are considered identical?
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For a group of 5 persons seated at a round table, how many unique arrangements are possible?
For a group of 5 persons seated at a round table, how many unique arrangements are possible?
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What is the key distinction when arranging persons in a circular manner as opposed to in a straight line?
What is the key distinction when arranging persons in a circular manner as opposed to in a straight line?
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If two beads are identical in a set of different colored beads, how does this affect the number of distinct necklace arrangements?
If two beads are identical in a set of different colored beads, how does this affect the number of distinct necklace arrangements?
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What is the main principle behind calculating the total arrangements when restrictions are applied?
What is the main principle behind calculating the total arrangements when restrictions are applied?
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How is a circular permutation defined in terms of ordinary permutations?
How is a circular permutation defined in terms of ordinary permutations?
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How many different kinds of single first class tickets may be printed if there are 50 stations on a railway line?
How many different kinds of single first class tickets may be printed if there are 50 stations on a railway line?
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How many six-digit numbers can be formed using the digits 9, 5, 3, 1, 7, 0 without repetition?
How many six-digit numbers can be formed using the digits 9, 5, 3, 1, 7, 0 without repetition?
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How many words can be formed with the letters of the word SUNDAY?
How many words can be formed with the letters of the word SUNDAY?
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How many different arrangements can be made with the letters of the word MONDAY?
How many different arrangements can be made with the letters of the word MONDAY?
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How many different arrangements can be made beginning with 'A' and ending in 'N' with the letters of the word ORIENTAL?
How many different arrangements can be made beginning with 'A' and ending in 'N' with the letters of the word ORIENTAL?
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In how many ways can a consonant and a vowel be chosen out of the letters of the word EQUATION?
In how many ways can a consonant and a vowel be chosen out of the letters of the word EQUATION?
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How many different words can be formed beginning with ‘T’ from the letters of the word TRIANGLE?
How many different words can be formed beginning with ‘T’ from the letters of the word TRIANGLE?
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In question No. (60), how many of the combinations will begin with ‘T’ and end with ‘E’?
In question No. (60), how many of the combinations will begin with ‘T’ and end with ‘E’?
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What is the correct calculation for the number of combinations of 52 cards taken 5 at a time?
What is the correct calculation for the number of combinations of 52 cards taken 5 at a time?
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How many triangles can be formed using 8 points in the plane where no three points are collinear?
How many triangles can be formed using 8 points in the plane where no three points are collinear?
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How many ways can a committee of 3 persons be formed from a selection of 12 individuals?
How many ways can a committee of 3 persons be formed from a selection of 12 individuals?
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What is the minimum number of Chartered Accountants required in a committee of 7 members that includes at least one from each profession?
What is the minimum number of Chartered Accountants required in a committee of 7 members that includes at least one from each profession?
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If 6 Chartered Accountants, 4 Economists, and 5 Cost Accountants are available, which method permits the maximum group of members while meeting all criteria?
If 6 Chartered Accountants, 4 Economists, and 5 Cost Accountants are available, which method permits the maximum group of members while meeting all criteria?
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What is the total number of ways to select committee members using Method 1?
What is the total number of ways to select committee members using Method 1?
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Which of the following represents the correct expression for choosing members in Method 2?
Which of the following represents the correct expression for choosing members in Method 2?
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In the context of forming combinations, which of the following statements is true?
In the context of forming combinations, which of the following statements is true?
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What is the number of permutations of n distinct objects taken r at a time when a specific object is excluded from all arrangements?
What is the number of permutations of n distinct objects taken r at a time when a specific object is excluded from all arrangements?
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In calculating the permutations of r objects when a specific object is always included, which formula is used?
In calculating the permutations of r objects when a specific object is always included, which formula is used?
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When arranging the letters of 'DRAUGHT' such that the vowels are not separated, how many distinct arrangements are possible?
When arranging the letters of 'DRAUGHT' such that the vowels are not separated, how many distinct arrangements are possible?
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What is the total number of arrangements of n books on a shelf such that two particular books are not together?
What is the total number of arrangements of n books on a shelf such that two particular books are not together?
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How are the two vowels 'A' and 'U' treated in the arrangements of 'DRAUGHT'?
How are the two vowels 'A' and 'U' treated in the arrangements of 'DRAUGHT'?
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What is the first step to determine the total number of arrangements when no restrictions apply to n books?
What is the first step to determine the total number of arrangements when no restrictions apply to n books?
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In the solution to keeping two particular books together on the shelf, what represents treating them as a single unit?
In the solution to keeping two particular books together on the shelf, what represents treating them as a single unit?
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What does the notation $2P2$ refer to in the context of the arrangement of two books on a shelf?
What does the notation $2P2$ refer to in the context of the arrangement of two books on a shelf?
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Study Notes
Permutations and Combinations
- The number of permutations of n distinct objects taken r at a time is nPr = n!/(n-r)!
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When a particular object is not taken in any arrangement:
- The number of permutations is (n-1)Pr
- This is because we are essentially arranging (n-1) objects in r places.
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When a particular object is always included in any arrangement:
- The number of permutations is r * (n-1)Pr-1.
- We can place the particular object in r different places.
- For each placement, we can arrange the remaining (r-1) objects with (n-1)Pr-1 permutations.
Circular Permutations
- Circular permutations are different than linear permutations since rotations of a permutation don't create new arrangements.
- The number of ways to arrange n objects in a circle is (n-1)!
- This is because any one object can be fixed as a reference point.
- The remaining (n-1) objects can be arranged linearly in (n-1)! ways.
Examples
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Example 1 (DRAUGHT):
- To arrange the letters of the word DRAUGHT without separating the vowels, we treat the two vowels as one unit.
- The vowels can be arranged in 2! ways (AU or UA).
- The six letters (5 consonants and the vowel unit) can be arranged in 6! ways.
- Therefore, the total number of arrangements is 2! * 6! = 1440 ways.
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Example 2 (Books on a shelf):
- The total number of ways to arrange n books is n!.
- To arrange n books so that two specific books are not together, we first find the arrangements where the two books are together.
- The two books can be arranged in 2! ways.
- We treat the two books as one unit and arrange them with the remaining (n-2) books, giving (n-1)! arrangements.
- Therefore, the number of arrangements where the two books are not together is n! - 2 * (n-1)! = (n-2) * (n-1)!
Applying the Fundamentals
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Example 3 (Committee of 3):
- We have 12 people, and we want to select a committee of 3.
- This is a combination, as the order of selection doesn't matter.
- The number of possible committees is 12C3 = 12!/(3! * 9!) = 220 ways.
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Example 4 (Committee with restrictions):
- A committee must have at least one member from each of three groups (Chartered Accountants, Economists, and Cost Accountants) and at least 3 Chartered Accountants.
- We need to consider different possible combinations of members from each group, making sure to meet the minimum requirements.
- For example, one possible combination is 3 Chartered Accountants, 2 Economists, and 2 Cost Accountants.
- Calculate the number of ways to choose members for each combination and then sum them across all possible combinations.
Mathematical Problems
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Problem 1: In how many ways can 7 boys sit at a round table so that two particular boys sit together?
- Treat the two boys as one unit.
- Arrange the 6 units (5 boys and the unit) around the circle.
- Arrange the two boys within their unit.
- Total possibilities: 5! *2! = 240 ways.
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Problem 2: The number of numbers between 100 and 1000 that can be formed using digits 1, 2, 3, 4, 5, 6, 7, 8:
- ** Hundreds place:** 7 choices ( any digit except 0).
- ** Tens and units place:** 7 choices each.
- Total: 7 * 7 * 7 = 343 numbers.
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Problem 3: How many words can be formed with the letters of the word
SUNDAY?- There are 6 letters. They can be arranged in 6! = 720 ways.
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Problem 4: The word
MONDAYcan be arranged in 6! = 720 ways. -
Problem 5: The word
ORIENTALhas 8 letters. There are 8! = 40,320 ways to arrange them.
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Description
Test your understanding of permutations and combinations, including both linear and circular arrangements. This quiz covers key concepts, formulas, and examples to reinforce your knowledge on counting techniques.
