Section 12.5 ­ Combinations & Permutations

Pre­Calculus

May 18, 2015

Independent Events - Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring

Examples: Picking two cards from a deck and replacing the first one.

In college a student has three different science classes and 2 different math classes to choose from. Selecting math course does not affect the selection of a science course.

Dependent Events - Two events, A and B are dependent if the fact that A occurs does affect the probability of B occurring

Examples:

Picking two cards from a deck without replacing the first one.

The order is which runners finish a race

Combinatorics - the branch of mathematics that studies the different possibilities for the arrangement of objects

Section 12.5 - Combinations & Permutations

I. Basic Counting Principal : suppose that one event can be chosen in p ways, and another independent event can be chosen in q ways. Then the two events can be chosen in p q ways.

1. Tom has decided to buy a new suit made of either wool or rayon. He has narrowed the color choices down to gray, blue, black, or tan. And matching ties to a paisley, power stripe, or solid. Are the choices independent or dependent? How many different selections of his suits and ties are possible?

Section 12.5 ­ Combinations & Permutations

Pre­Calculus

May 18, 2015

2. The U.S. Postal Service uses 5-digit ZIP codes to route letters and packages to their destination.

a.) How many ZIP codes are possible if the numbers 0 through 9 are used for each of the 5 digits?

b.) Suppose that when the first digit is 0, the second, third, and fourth digits cannot be 0. How many 5-digit ZIP codes are possible if the first digit is 0?

II. Permutation: arrangement of objects in a certain orderorder matters.

The symbol P(n, n) denotes the number of permutations of n objects taken all at once

This is defined as: P(n, n) = n!Example: How many ways are there to display 5 books?

The symbol P(n, r) denotes the number of permutations of n objects taken r at a time

This is defined as: P(n, r) = n! (n - r)!

Example: If there are 5 books, how many different ways can I display 3 of them.

Should be called a permutation lock

Section 12.5 ­ Combinations & Permutations

Pre­Calculus

May 18, 2015

2. The board of directors of a major corporation is composed of 10 members.

- How many different ways can the 10 members sit at a conference table?

- How many ways can they elect a president, vice president, secretary and treasurer? Keep in mind that one person cannot hold more than one office.

3. A high school honor society is composed of 7 students.

- How many different ways can they be arranged for a picture?

- In how many ways can they select a president, vice president, and secretary? Assuming that one person cannot hold more than one office.

Section 12.5 ­ Combinations & Permutations

Pre­Calculus

May 18, 2015

III. Combination: when the arrangement of objects can be in any order - order DOES NOT matter .

The symbol C(n, r) denotes the number of combinations of n objects taken r at a time

This is defined as: C(n, r) = n! (n - r)! r!

See...I told you "combination" lock is a misnomer

* Remember that !, means factorial...So 5! = 5 4 3 2 1

1. The national art gallery in Washington wants to select 4 paintings from a possible 20. How many groups of 4 paintings can be chosen?

- Does order matter? So is it a permutation or combination?

- How many groups of 4 paintings can be chosen?

Section 12.5 ­ Combinations & Permutations

Pre­Calculus

May 18, 2015

2. Every year a popular magazine picks the top 15 rated TV movies from the previous year. A network wants to select 3 of these 15 movies to show.

- Does order matter? So is it a permutation or combination?

- How many groups of 3 movies can be chosen?