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?