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.

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Section 15 4 day 1 permutations with repetition circular permutations

Section 15.4 Day 1:Permutations with Repetition/Circular Permutations

Pre-calculus


Learning targets
Learning targets

  • Recognize permutations with repetition

  • Solve problems that involve circular permutations


Problem 1
Problem 1

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

  • Write down all the different permutations of the word MOM


Problem 11
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 12
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
# 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
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
Example 2

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


Example 3
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
Circular Permutations

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

  • For example, people sitting around at a table.


Circular permutations1
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 permutations2
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
Example 4

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

    9!


Example 5
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
Homework

  • Textbook Page 585-586

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


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