Warm-Up Problem
Can you predict which offers more choices for license plates?
Choice A: a plate with three different letters of the alphabet in any
orderChoice B: a plate with four different
nonzero digits in any order.
By the end of class, you should be able to successfully answer this question.
Example 1
Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of paper is drawn from the box, its number is written down, and the piece of paper is replaced in the box.
A second piece of paper is drawn from the box, and its number is written down. If the two numbers are added together, how many different ways can a total of 12 be obtained?
Example 2
Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of paper is drawn from the box and its number is written down.
A second piece of paper is drawn from the box, and its number is written down.
If the two numbers are added together, how many different ways can a total of 12 be obtained?
Example 3
You have three books that you want to place on a shelf.
How many different ways can you arrange the books on the shelf?
The Counting Principle
Let E1 and E2 be two events. The first event E1 can occur m different ways.
The second event E2 can occur n different ways.
The number of ways that the two events can occur is m times n.
Example 3 again.
You have three books that you want to place on a shelf.
How many different ways can you arrange the books on the shelf?
6 different ways to place 3 books on a shelf.
Example 4
You have five books and you want to place three of them on a shelf.
How many different ways can you arrange three of the five books on the shelf?
60 different ways
This is the number of different arrangements or permutations of 5 things taken 3 at a time.
This can be written as P(5,3).
Remember: Permutations of n Elements taken r at a time is found by
n r
n!P n, r P
n r !
Key Words: arrangements, schedule, order
Example 5
Eight horses are running in a race. In how many different ways can these horses come in first, second, and third? Assume there are no ties.
Using the Counting Principle: 8(7)(6)=336.
8 38!
P 3368 3 !
Using Permutations:
Example 6Your teacher gives you a list of three books. You must choose two of the books on the list to read.
How many different pairings can you choose from?
Combinations
When you choose r elements from a large set of n elements and ORDER DOES NOT MATTER,
You are finding the number of combinations of n elements taken r at a time.
n r
n n!C n, r C
r r ! n r !
Key Words: group, committee, selection, sample
Example 7
How many different committees of 3 people can be chosen from a group of 8 people?
Since order does not matter, use combinations
8 8!3 3! 8 3 !
56 committees
Warm-Up Problem
Can you predict which offers more choices for license plates?
Choice A: a plate with three different letters of the alphabet in any
orderChoice B: a plate with four different
nonzero digits in any order.
Alphabet – 15,600 Numbers - 3024