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

2. 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?

3. 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

4. 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?

5. 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?

6. 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

7. 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

8. 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?

9. 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. 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)!

11. 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?

12. 10.4 – The Fundamental Counting Principle and Permutations
HOMEWORK
10.4 Practice A Worksheet