Name: ____________________ Pre- Calculus 12 H. Date: _____________

Chapter 11 – Permutations, Combinations, and the Binomial Theorem

11.1– Permutations

Objectives To introduce the concept of a permutation.

To find the number of permutations of n distinct objects taken r at a time.

Investigate

1. Two letters, A and B, can be written in two different orders, AB and BA.

These are permutations of A and B.

a) List all of the permutations of 3 letters A, B, and C.

How many permutations are there?

b) List all of the permutations of 4 letters A, B, C, and D.

How many permutations are there?

c) Predict the number of permutations of 5 letters A, B, C, D, and E.

2. Instead of arranging letters in order, we can arrange objects if they are all different.

a) How many different ways can 5 people be arranged in a line?

b) How many different ways can 5 different books be arranged on a shelf?

c) How many permutations can be formed using all of the letters of the word COMPUTE?

Example 1: Jason is to come up with a name for his baby using J A S O N. If he is to re-arrange all five

letters in his name, determine the number of names he can come up with (including JASON).

PERMUTATION

When a number of objects are to be arranged, and the order of the selection is important

*From last class… Factorial Notation: n! =

Example 2: How many permutations can be

formed using all the letters of the word

CHEMISTRY?

Example 3: In how many different ways can 14

horses finish a horse race?

Example 4: How many different “words” can be formed using all the letters of M I S S I S S I P P I ?

Rule: Consider n objects all at once. If “a” of them are alike (identical) of one kind, “b” of them are alike

of another kind, and so on… the number of permutations for choosing all n objects is:

!...!!

!

cba

n

Example 5: On a 20-question True/False test, 8 answers are True and 12 answers are False. How many

possible answer keys are there?

Example 6: A group of 20 is going for the Sun Run. Find the number of permutations for each case.

a) All of them finish the race? b) The runners place 1st, 2nd, and 3rd?

Example 7: WVSS has a group of 200 going for a marathon.

In how many ways could they finish 1st through 20th?

Special type of notation for some situations

The number of permutations of n distinct objects taken r at a time is:

!

( )!

n

n r = n Pr

MATH PRB 2: n Pr

Example 8: Consider the word SMART. How many of the following permutations are there?

a) 2-letter permutations? b) 3-letter permutations?

Example 9: Jay and Ivy invited 3 other people to sit on their bench. In how many ways can these five

people be seated on this bench if the following conditions apply?

a) No restrictions b) Jay is on the left and Ivy is on the right.

Example 10: Solve for n: a) 422 Pn b) 1206 nP

Example 11: Five people (A, B, C, D, and E) are seated on a bench. In how many ways can they be

arranged if:

a) E is seated in the middle?

b) A and B must be seated together?

c) A and B cannot sit together?

Example 12: Moving only right and/or down, how many paths are there to go from A to B?

a) b)

Example 13: How many 4-letter “words” can be formed from the word GROUPS if there is/are:

a) No restrictions

b) No vowels

c) Begin and end in a vowel

Assignment: p. 524 #1ac, 5-8, 10-12