Section 12.8 The Counting Principle and Permutations

What You Will Learn The Counting Principle Permutations

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.

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

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

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

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

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)

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.

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.

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

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?

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

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

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.