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Section 15.4 Day 1: Permutations with Repetition/Circular PermutationsPowerPoint Presentation

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

MOP M1OM2 Notice that MOM

MPO M1M20 gives only 3 types

OMP OM1M2 if the M’s are the

OPM OM2M1 same and not different

PMO M2M10 MOM, MMO, OMM

POM M2OM1

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
- First choose the “leader”: Leader , ____, _____, _____
- 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

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