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Permutations and Combinations

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PRE-ALGEBRA LESSON 12-6

Bianca’s family needs to choose exterior paint for their new house. The wall colors are white, green, and beige. The trim colors are white, green, blue, and cocoa. How many combinations of wall color and trim are possible?

12

12-6

PRE-ALGEBRA LESSON 12-6

(For help, go to Lesson 12-4.)

Use the Counting Principle to find the number of outcomes.

1.Roll 2 number cubes.

2.Choose three different letters.

3.Select a month and a day of the week.

4.Toss a coin 4 times.

Check Skills You’ll Need

12-6

PRE-ALGEBRA LESSON 12-6

Solutions

1.first numbersecond number

6 outcomes6 outcomes

66= 36

2.first lettersecond letterthird letter

(different from 1st)(different from 1st & 2nd)26 choices25 choices24 choices

26 25 24

= 15,600

12-6

PRE-ALGEBRA LESSON 12-6

Solutions (continued)

3.monthday

12 choices7 choices

12 7= 84

4.first tosssecond tossthird tossfourth toss

2 outcomes2 outcomes2 outcomes2 outcomes

2 2 2 2

= 16

12-6

1st letter

5 choices

5

2nd letter

4 choices

4

3rd letter

3 choices

3

4th letter

2 choices

2

5th letter

1 choice

1

= 120

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PRE-ALGEBRA LESSON 12-6

Find the number of permutations possible for the letters H, O, M, E, and S.

There are 120 permutations of the letters H, O, M, E, and S.

Quick Check

12-6

7 studentsChoose 3.

7P3

PRE-ALGEBRA LESSON 12-6

In how many ways can you line up 3 students chosen from 7 students for a photograph?

= 7 • 6 • 5 = 210Simplify

You can line up 3 students from 7 in 210 ways.

Quick Check

12-6

State Area (mi2)

Alabama

Colorado

Maine

Oregon

Texas

50,750

103,729

30,865

96,003

261,914

PRE-ALGEBRA LESSON 12-6

In how many ways can you choose two states from the table when you write reports about the areas of states?

Make an organized list of all the combinations.

12-6

PRE-ALGEBRA LESSON 12-6

(continued)

AL, COAL, MEAL, ORAL, TXUse abbreviations of each

CO, MECO, ORCO, TXstate’s name. First, list all

ME, ORME, TXpairs containing Alabama.

OR, TXContinue until every pair

of states is listed.

There are ten ways to choose two states from a list of five.

Quick Check

12-6

9 toppingsChoose 5.

9C5

9P5

5P5

=

9 • 81 • 7 • 62 • 51

51 • 41 • 31 • 21 • 1

= = 126 Simplify.

PRE-ALGEBRA LESSON 12-6

How many different pizzas can you make if you can choose exactly 5 toppings from 9 that are available?

You can make 126 different pizzas.

Quick Check

12-6

PRE-ALGEBRA LESSON 12-6

Tell which type of arrangement—permutations or combinations—each problem involves. Explain.

a. How many different groups of three vegetables could you choose from

six different vegetables?

b. In how many different orders can you play 4 DVDs?

Combinations; the order of the vegetables selected does not matter.

Permutations; the order in which you play the DVDs matters.

Quick Check

12-6

PRE-ALGEBRA LESSON 12-6

Find the number of permutations or combinations.

1.Find the number of permutations of the last four digits in the phone number 555–1234.

2.In how many ways can you arrange six out of eight books on a shelf?

3.In how many ways can you choose three different items from a menu containing seven items?

Does each problem involve permutations or combinations?

4.How many different groups of four ice cream toppings can you choose from sixteen toppings?

5.In how many different ways can five students line up?

24

20,610

35

combinations

permutations

12-6