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 differentPMO 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

• 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

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

EXAMPLE 4

• 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?

• Treat each couple as one item, so we have 4 items• We have to remember that the husbands/wives can switch seats,

and this is a different permutation

(3!)(

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!

HOMEWORK

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