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 MOPM 1 OM 2 Notice that MOM MPOM 1 M 2 0gives only 3 types OMPOM 1 M 2 if the M’s are the OPMOM 2 M 1 same and not different PMOM 2 M 1 0MOM, MMO, OMM POMM 2 OM 1

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

7. EXAMPLE 1

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 How can we decide what makes a circular permutation? Notice the pictures are the same permutations because it follows the order ABCD regardless of which letter is on top. To have different circular permutations, we could have ABCD, DABC, CDAB, BCDA

12. EXAMPLE 3 How many circular permutations are possible when seating four people around a table? We can deconstruct the circular permutations into a linear permutation Choose a “leader” and then permute the rest (“A”, __, __, __) If n distinct objects are arranged around a circle, then there are (n – 1)! circular permutations of the n objects. Thus, there are 6 different ways

13. EXAMPLE 4

14. EXAMPLE 5 How many ways are there to arrange 3 women and 3 men alternating at a table? We don’t need to use a circular permutation on the men since each situation they sit is different for the scenario. 2!3!

15. HOMEWORK Textbook Page 585-586 (Written Exercises) #1-5odd, 9, 11, 12