## 18 Questions

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?

720

How many ways can the numbers 2345678 be rearranged if the number must begin with exactly two odd digits?

3600

What is the value of $n$ in the combination formula $nCr = \frac{n!}{r!(n-r)!}$ if $r = 3$ and $nCr = 10$?

6

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?

240

What is the number of combinations of 5 items chosen 2 at a time?

15

How many ways can the letters in the word 'ACTIVE' be arranged if C and E must never be together?

60

How many ways can you arrange 5 different books on a shelf if the first and last books are already chosen?

4! × 2!

How many committees of 3 can be formed from 10 people if one person refuses to work with another person?

10C3 - 9C2

If a password consists of 3 uppercase letters and 4 digits, how many different passwords are possible if repetition is allowed?

26^3 × 10^4

How many ways can you arrange 5 people in a row if 2 people must always be next to each other?

4! × 2!

How many different ways can you select 4 books from a shelf of 10 books if the first and last books are already chosen?

8C2

If a license plate consists of 3 uppercase letters and 2 digits, how many different license plates are possible if repetition is not allowed?

26P3 × 10P2

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?

40 x 3!

If a box contains 12 identical pens, and 8 identical pencils, how many ways are there to choose 4 items from the box?

495

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?

5

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?

n(A ∪ B ∪ C ∪ D ∪ E)

If a password consists of 2 uppercase letters, and there are 26 uppercase letters, how many possible passwords are there?

26^2

If a committee consists of 3 people chosen from a group of 10 people, how many ways are there to choose the committee?

10 choose 3

## 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.

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!

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