Permutations and Combinations

Introduction

Permutations and combinations are fundamental concepts in probability and statistics, used to count the number of ways objects can be arranged or selected. Understanding when to use each is crucial for solving problems involving arrangements and selections.

Permutations

Definition and Key Concepts

• Permutation: An arrangement of n different objects in a specific order. The order of arrangement is important.

• The goal is to find the number of ways to select r objects out of n objects, where order matters.

Permutation Rule 1: Arranging All Objects

• The number of ways to arrange all n different objects is given by n factorial (n!).

• Formula:

• By definition: and

Examples:

Application Example: If a sports anchor wants to rank all 5 teams, the number of ways is .

Application Example: Seating 6 students in a row of 6 desks: ways.

Permutation Rule 2: Arranging r Objects from n

• Arranging r objects out of n in a specific order is called a permutation of n objects taken r at a time.

• Notation: or

• Formula:

• Conditions: ; both n and r are whole numbers.

Examples:

Application Example: Ranking the top 3 teams out of 5: ways.

Application Example: Seating 4 out of 6 students: ways.

Application Example: Selecting a chairperson and assistant from 7 scientists: ways.

Application Example: Arranging 3 boxcars from 8: ways.

Application Example: Arranging 4 photos from 9: ways.

Permutation Rule 3: Permutations with Indistinguishable Objects

• Used when some objects are identical.

• Formula: , where

Example: Arranging 3 yellow, 5 blue, 2 red, and 1 green book (total 11 books):

• ways

Application Example: Permutations of the letters in "SEEM":

Application Example: Permutations of the letters in "Mississippi":

Application Example: Arranging 6 Algebra, 2 Trigonometry, and 7 Geometry books: ways

Combinations

Definition and Key Concepts

• Combination: A selection of r objects from n objects where order does not matter.

• Notation: or or

• Formula:

• Conditions:

Examples:

Application Example: Choosing 4 books from 12: combinations.

Application Example: Choosing 5 freshmen from 8: combinations.

Permutations vs. Combinations

• Permutation: Use when order is important.

• Combination: Use when order is not important.

Using Combinations to Find Probabilities

Finding Probabilities with Combinations

Combinations are often used to calculate probabilities when the order of selection does not matter.

• Example: Three pool balls are randomly chosen from 15. What is the probability that the balls are numbered 5, 7, and 9?

• Step 1: Find the number of ways to choose the specific balls (numerator): 1 way.

• Step 2: Find the total number of ways to choose any 3 balls from 15 (denominator):

• Step 3: Probability =

Summary Table: Permutations and Combinations

Additional info: The images provided (image_1, image_2, image_3) do not contain any diagrams, formulas, or visual aids directly relevant to the explanation of permutations or combinations. Therefore, no images are included in these study notes.