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Questions and Answers
What is the total number of permutations of the letters in the word 'KITCHEN' if it must start with a consonant and end with a vowel?
How many ways can the numbers 2345678 be rearranged if the number must begin with exactly two odd digits?
What is the value of $n$ in the combination formula $nCr = \frac{n!}{r!(n-r)!}$ if $r = 3$ and $nCr = 10$?
How many ways can Brittany, Geoffrey, Jonathan, Kyle, Laura, and Stephanie stand in a line if boys and girls alternate with a boy starting the line?
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What is the number of combinations of 5 items chosen 2 at a time?
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How many ways can the letters in the word 'ACTIVE' be arranged if C and E must never be together?
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How many ways can you arrange 5 different books on a shelf if the first and last books are already chosen?
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How many committees of 3 can be formed from 10 people if one person refuses to work with another person?
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If a password consists of 3 uppercase letters and 4 digits, how many different passwords are possible if repetition is allowed?
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How many ways can you arrange 5 people in a row if 2 people must always be next to each other?
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How many different ways can you select 4 books from a shelf of 10 books if the first and last books are already chosen?
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If a license plate consists of 3 uppercase letters and 2 digits, how many different license plates are possible if repetition is not allowed?
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If a committee consists of three members, with two members selected from a group of five men and one member selected from a group of four women, how many different committees can be formed?
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If a box contains 12 identical pens, and 8 identical pencils, how many ways are there to choose 4 items from the box?
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If a student can choose to play either tennis or basketball, and there are 3 tennis courts and 2 basketball courts, how many choices does the student have?
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If there are 5 events, A, B, C, D, and E, and events A and B are disjoint, and events C and D are disjoint, how many ways are there for at least one of the events to occur?
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If a password consists of 2 uppercase letters, and there are 26 uppercase letters, how many possible passwords are there?
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If a committee consists of 3 people chosen from a group of 10 people, how many ways are there to choose the committee?
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Study Notes
Conditional Permutations
- To find the number of permutations with conditions, identify the restrictions and calculate the possible arrangements.
- Example: In the word "ACTIVE", if C & E must always be together, there are 23 possible arrangements.
- Example: In the word "ACTIVE", if C & E must never be together, there are 24 possible arrangements.
Combinations
- A combination is an arrangement of items where order does not matter.
- The formula to find the number of combinations of items chosen at a time is 𝑛! / (𝑟!(𝑛-𝑟)!) where 0≤𝑟 ≤𝑛.
- Example: How many 4-letter words can be created if repetitions are not allowed?
- Example: How many three-letter words can be made from the letters of the word KEYBOARD?
Combinations Examples
- If there are 35 songs and you want to make a mix CD with 17 songs, there are many ways to arrange them.
- If there are six different colored balls in a box, and you pull them out one at a time, there are many ways to pull out 4 balls.
- A committee is to be formed with a president, a vice-president, and a treasurer from 20 people, and there are many different committees possible.
Permutations with Conditions
- Permutations with specific positions require analyzing how many possible ways each space can be filled.
- Example: How many numbers can be made from rearranging 2345678 if the number must begin with two odd digits?
- Permutations with always together conditions require treating the objects together as 1 and determining the number of arrangements, and then finding the number of internal arrangements.
Lesson Outcomes
- Apply the "Sum Rule" Principle and "Product Rule" Principle.
- Distinguish between "Permutations and Combinations".
- Calculate Permutations and Combinations.
Combinatorial Analysis
- Combinatorial Analysis includes the study of permutations and combinations.
- It is concerned with determining the number of logical possibilities of some event without necessarily identifying every case.
Sum Rule Principle
- If some event can occur in 𝑚 ways and a second event can occur in 𝑛 ways, and suppose both events cannot occur simultaneously, then it can occur in 𝑚 + 𝑛 ways.
- Example: Computing faculty must choose either a student or a faculty member as a representative for a university committee, and there are 27 faculty members and 83 CS majors.
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Description
Test your knowledge of combinatorics and permutations with these questions on arranging words, songs, and committee members. Practice your problem-solving skills with these challenging scenarios!
