1. Section 12.8 The Counting Principle and Permutations

2. What You Will Learn
The Counting Principle
Permutations

3. Counting Principle
If a first experiment can be performed in M distinct ways and a second experiment can be performed in N distinct ways, then the two experiments in that specific order can be performed in M • N distinct ways.

4. Example 1: Counting Principle: Passwords
A password used to gain access to a computer account is to consist of two lower case letters followed by four digits. Determine how many different passwords are possible if a) repetition of letters and digits is permitted.
Solution
26•26•10•10•10•10 = 6,760,000 different possible passwords

5. Example 1: Counting Principle: Passwords
b) repetition of letters and digits is not permitted.
Solution
26•25•10•9•8•7 = 3,276,000 different possible passwords

6. Example 1: Counting Principle: Passwords
c) the first letter must be a vowel (a, e, i, o, u) and the first digit cannot be a 0, and repetition of letters and digits is not permitted.
Solution
5•25•9•9•8•7 = 567,000 different possible passwords

7. Permutations
A permutation is any ordered arrangement of a given set of objects.

8. Number of Permutations
The number of permutations of n distinct items is n factorial, symbolized n!, where
n! = n(n – 1)(n – 2) • • • (3)(2)(1)

9. Example 3: Cell Phones
In how many different ways can six different cell phones be arranged on top of one another?
Solution
6! = 6 • 5 • 4 • 3 • 2 • 1 = 720
The 6 cell phones can be arranged in 720 different ways.

10. Example 4: Permutation of Three Out of Five Letters
Consider the five letters a, b, c, d, e. In how many distinct ways can three letters be selected and arranged if repetition is not allowed?
Solution
5 • 4 • 3 = 60
Thus, there are 60 different possible ordered arrangements, or permutations.

11. Permutation Formula
The number of permutations possible when r objects are selected from n objects is found by the permutation formula

12. Example 5: Using the Permutation Formula
You are among eight people forming a skiing club. Collectively, you decide to put each person’s name in a hat and to randomly select a president, a vice president, and a secretary. How many different arrangements or permutations of officers are possible?

13. Example 5: Using the Permutation Formula
Solution
n = 8, r = 3

14. Permutations of Duplicate Objects
The number of distinct permutations of n objects where n1 of the objects are identical, n2 of the objects are identical, …, nr of the objects are identical is found by the formula

15. Example 7: Duplicate Letters
In how many different ways can the letters of the word “TALLAHASSEE” be arranged?
Solution
Of the 11 letters, 3 are A’s, 2 are S’s, 2 are L’s, and 2 are E’s.