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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. 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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  1. Section 15.4 Day 1:Permutations with Repetition/Circular Permutations Pre-calculus

  2. Learning targets • Recognize permutations with repetition • Solve problems that involve circular permutations

  3. Problem 1 • Write down all the different permutations of the word MOP. • Write down all the different permutations of the word MOM

  4. 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

  5. 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.

  6. # 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:

  7. 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!

  8. Example 2 • How many distinguishable permutations are there of the letters of MASSACHUSETTS?

  9. 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?

  10. Circular Permutations • In addition to linear permutations, there are also circular permutations. • For example, people sitting around at a table.

  11. 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.

  12. 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)!

  13. Example 4 • How many ways are there to arrange 5 boys and 5 girls? 9!

  14. 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!)(

  15. Homework • Textbook Page 585-586 (Written Exercises) #1-5odd, 9, 11

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