# Combinatorics and Permutations Quiz

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## 18 Questions

720

3600

6

240

15

60

4! × 2!

10C3 - 9C2

26^3 × 10^4

4! × 2!

8C2

26P3 × 10P2

40 x 3!

495

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)

26^2

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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