# Section 15.4 Day 1: Permutations with Repetition/Circular Permutations - PowerPoint PPT Presentation

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Section 15.4 Day 1: Permutations with Repetition/Circular Permutations. Pre-calculus. Learning targets. Recognize permutations with repetition Solve problems that involve circular permutations. Problem 1. Write down all the different permutations of the word MOP.

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Section 15.4 Day 1: Permutations with Repetition/Circular Permutations

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## Section 15.4 Day 1:Permutations with Repetition/Circular Permutations

Pre-calculus

### Learning targets

• Recognize permutations with repetition

• Solve problems that involve circular permutations

### Problem 1

• Write down all the different permutations of the word MOP.

• Write down all the different permutations of the word MOM

### Problem 1

MOPM1OM2Notice that MOM

MPOM1M20gives only 3 types

OMPOM1M2if the M’s are the

OPMOM2M1same and not different

PMOM2M10MOM, MMO, OMM

POMM2OM1

### Problem 1

• Thus, with MOP and MOM there are 3! = 6 total permutations.

• However, if we are looking for DISTINGUISHABLE permutations, MOP would still have 6 but MOM would only have 3.

### # of Permutations of objects not all different

• Let S be a set of n elements of k different types.

• Let be the number of elements of type 1

• Let be the number of elements of type 2

• Let be the number of elements of type k

• Then the number of distinguishable permutations of the n elements is:

### Example 1

• How many distinguishable permutations are there of the letters MOM?

• n = 3

• = 2 M’s

• = 1 O

• This matches our observations from before!

### Example 2

• How many distinguishable permutations are there of the letters of MASSACHUSETTS?

### Example 3

• The grid shown at the right represents the streets of a city. A person at point X is going to walk to point Y by always traveling south or east. How many routes from X to Y are possible?

### Circular Permutations

• In addition to linear permutations, there are also circular permutations.

• For example, people sitting around at a table.

### Circular Permutations

• Question: How can we decide what makes a circular permutation?

• Consider:The pictures below are the same permutations because it follows the same order regardless of which color starts on top.

### Circular Permutations

• To determine the number of circular permutations, wecan deconstruct the circular permutations into a linear permutation

• Then permute the remaining spaces.

• This always ends up as (n-1)!

### Example 4

• How many ways are there to arrange 5 boys and 5 girls?

9!

### Example 5

• How many ways are there to seat 4 husbands and 4 wives around a dining table such that each husband is next to his wife?

(3!)(

### Homework

• Textbook Page 585-586

(Written Exercises) #1-5odd, 9, 11