Chapter 10 – Data Analysis and Probability 10.4 – The Fundamental Counting Principle and Permutations

10.4 – The Fundamental Counting Principle and Permutations If your lunch options are chicken, pasta or fish for the main dish and soup or salad for the side dish, how many different lunch choices do you have?

10.4 – The Fundamental Counting Principle and Permutations Two Events If one event can occur in m ways and another event can occur in n ways, then the number of ways that BOTH events can occur is m · n Three Events If one event can occur in m ways, a second event can occur in n ways, and a third event can occur in p ways, then the number of ways that ALL THREE events can occur is m · n · p The principle also applies to four or more events

10.4 – The Fundamental Counting Principle and Permutations Example 1 At a blood drive, blood can be labeled one of four types (A, B, AB, or O) one of two Rh factors (+ or -), and one of two genders (F or M). How many different ways can blood be labeled?

10.4 – The Fundamental Counting Principle and Permutations Example 2 Suppose a bicycle license plate is 2 letters followed by 4 digits. How many license plates are possible if letters and digits can be repeated? How many license plates are possible if letters and digits cannot be repeated?

10.4 – The Fundamental Counting Principle and Permutations Permutation – an ordering of a set of objects is a permutation of the objects. By the fundamental counting principle, there are 6 permutations (3 · 2 · 1) of 3 objects. Ex. A, B, C ABC, ACB, BAC, BCA, CAB, CBA

10.4 – The Fundamental Counting Principle and Permutations You can also write the expression 3 · 2 · 1 as 3!. Factorial (!) – the number multiplied by one less each time until you get to one Ex. 4! = 4 · 3 · 2 · 1 Ex. 5! = 5 · 4 · 3 · 2 · 1 Ex. 6! = 6 · 5 · 4 · 3 · 2 · 1

10.4 – The Fundamental Counting Principle and Permutations Example 3 A television news director has 8 news stories to present on the evening news. How many different ways can the stories be presented? If only 3 of the stories will be presented, how many possible ways can a lead story, a second story, and a closing story be presented?

10.4 – The Fundamental Counting Principle and Permutations In the previous example, there were 3 news stories chosen from 8 total news stories. This is a permutation of 8 objects taken 3 at a time (8P3). 8P3 = 8 · 7 · 6 = 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 5 · 4 · 3 · 2 · 1

10.4 – The Fundamental Counting Principle and Permutations Permutations of n Objects Taken r at a Time The number of permutations of n objects taken r at a time is denoted by nPr and is given by the following formula: n P r = n! . _(n – r)!

10.4 – The Fundamental Counting Principle and Permutations Example 4 Your grocery shopping cart has 8 items. In how many orders can the checkout clerk scan 5 of the items? In how many orders can the clerk scan all 8 of the items?

10.4 – The Fundamental Counting Principle and Permutations HOMEWORK 10.4 Practice A Worksheet