Counting The Fundamental Counting Principle

Fundamental Counting Principle If a series of “n” decisions must be made, and if the first decision can be made in x 1 ways, the second decision can be made in x 2 ways, the third decision can be made in x 3 ways and so on until the nth decision can be made in x n ways, then the number of different ways in which all of the “n” decisions could collectively be made is: x 1  x 2  x 3  …  x n (the product of the number of ways in which each of the individual decisions can be made

The Fundamental Counting Principle Example 1: How many different outfits can Al wear if he has a choice of 5 dress shirts, 3 ties, 2 pairs of dress pants, and 2 pairs of shoes? Assume he selects one of each item. Solution:Al needs to make 4 decisions: 1) What shirt? 2) What tie? 3) What pair of pants? 4) What pair of shoes?

The Fundamental Counting Principle The first decision can be made in 5 ways, the second decision can be made in 3 ways, the third and fourth decision can be made in 2 ways. Thus, the number of ways in which all four decisions can be made is: 5  3  2  2 = 60 Therefore, Al has 60 different outfits in which he can appear.

The Fundamental Counting Principle Example 2: In how any orders can 5 different books be arranged on a shelf if the books are right side up and the spine faces outwards? Solution: There are 5 decisions to be made. We must first decide what book will occupy the leftmost position. There are five books from which to choose for this position. Next, we must decide which book to place in the second position from the left. Any of the remaining four books can be placed there. This continues until there is only one choice for the book on the extreme right.

The Fundamental Counting Principle A factorial symbol, an exclamation mark “!”, is often used to shorten the writing of a product of integers such as this. 5! = 5  4  3  2  1 = 120 Therefore, there are 120 different ways of arranging the five books. Factorial  Notation

Homework Do # 1 – 4, 7, 13, and 14 on page 190 from section 6.1 for Thursday May 28 th