Download Combinations and Permutations

Combinations and Permutations
An arrangement where order is important is called a permutation.
Example: Mario, Sandy, Fred, and Shanna are running for the offices of president, secretary and
treasurer. In how many ways can these offices be filled?
Any of the 4 people can fill the president's position. Once the president has been chosen, there
are 3 people left to fill the secretary's position. That leaves 2 people to fill the treasurer's
position. The symbol 4 P3 represents the number of permutations of 4 things taken 3 at a time.
4 P3 = 4 x 3 x 2 = 24. The offices can be filled 24 ways.
An arrangement where order is not important is called combination.
Example: Charles has four coins in his pocket and pulls out three at one time. How many
different amounts can he get?
The four coins are a penny, a nickel, a dime, and a quarter. There are 4 x 3 x 2, or 24 different
outcomes, but some are the same. To find the number of different combinations, divide the
number of permutations 4 P3 =24 , by the number of different ways three items can be arranged.
4 x3x 2 24
4 P3
4 . Thus 4 different amounts can be chosen.
4 C3
3!
3x 2 x1 6
Determine if the situation represents a permutation or a combination, and then solve:
1. In how many ways can five books be arranged on a book-shelf? (Order is important.)
2. In how many ways can three student-council members be elected from five candidates?
3. Seven students line up to sharpen their pencils.
4. A DJ will play three CD choices from the 5 requests.
5. How many ways can the letters of RULES be arranged to make different “words”?
6. How many ways can a club of 6 members choose a 3-person committee?
Combination and Permutation Homework
1. Tell whether order is important or not important for each event.
a. permutation
b. combination
c. making up a 3-digit password
d. arranging encyclopedias on a bookshelf
e. picking friends for a sleep over
f. choosing a committee to discuss drug use in college
2. List all the permutations of 2 letters, chosen from A, B, C, D, and E.
3. List all the combinations of 2 letters, chosen from P, Q, R, S, and T.
4. How many ways can 5 children be seated in a row for pictures? Explain how you solved.
5. There are 7 different TV dinners in the freezer. In how many ways could Suzy eat the
dinners if she eats one dinner per night Monday through Friday?
6. Suzy goes shopping on Saturday and picks 5 different flavors from the 7 choices for TV
dinners. In how many ways can she pick 5 different dinners?
7. In some card games with a deck of 52 cards, a “hand” is made up of five cards. How
many different hands are possible?
8. In a lottery game, you pick six two-digit numbers from 00 through 99. You are the big
winner if all six of your choices match the six that come up in the lottery. What is the
probability that you are the big winner?
9.
How many 5-digit zip codes are possible if digits can be repeated?
10. How many different ways can 7 different video games be arranged on a self?
Extension: Explain the connection of how we solve these problems to the following formulas.
a) n Pr
n!
(n r )!
b) n C r
n!
(n r )!r!