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*11-1 Permutations and Combinations Related Word: combination (noun) Definition: A permutation is an*

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Vocabulary

Chapter 11 278

11-1

Review

1. Circle each word whose meaning is similar to that of fundamental.

basic essential extra necessary secondary unimportant

Vocabulary Builder

permutation (noun) pur myoo TAY shun

Related Word: combination (noun)

Definition: A permutation is an arrangement of items in a particular order.

Main Idea: When you put a set of objects in a certain order, you make a permutation.

Use Your Vocabulary

2. Write a permutation of each set.

., >, ,, < A, W, 6, 7, 2, E, V y, o, U, R

3. Write all possible permutations of the numbers 5, 9, and 8.

4. There are permutations of the numbers 5, 9, and 8.

set of numbers 5, 7, 11, 8

permutation 11, 7, 8, 5

Using the Fundamental Counting Principle

Got It? In 1966, one type of Maryland license plate had two letters followed by four digits. How many of this type of plate were possible?

5. Multiple Choice Which set of letters and digits gives a possible 1966 Maryland license plate?

TV 432 RC 2301 KPH 621 OMNQ 23

Permutations and Combinations

Problem 1

Sample answers are given.

., , , W, A, V, E, 6, 7, 2 U, o, y, R

5, 9, 8; 5, 8, 9; 8, 5, 9; 8, 9, 5; 9, 8, 5; 9, 5, 8

6

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279 Lesson 11-1

6. There are possible letters for each letter in the license plate.

There are possible digits for each digit.

7. Use the Fundamental Counting Principle to find the number of possible license plates. Complete the expression.

26 ? ? ? ? ?

8. Circle the number of possible 1966 Maryland license plates.

26,000 45,697,600 6,760,000

Using factorial notation, you can write 3 ? 2 ? 1 as 3!, read “three factorial.” For any positive integer n, n factorial is n! 5 n(n 2 1) ? c ? 3 ? 2 ? 1. Th e zero factorial is 0! 5 1.

Finding the Number of Permutations of n Items

Got It? In how many ways can you arrange 8 shirts on hangers in a closet?

9. Complete the model below.

1st shirt

Write

Relate

68

2nd shirt

7

3rd shirt

5

4th shirt

4

5th shirt

3

6th shirt

2

7th shirt

1

8th shirt

number of ways to arrange the

10. The total number of permutations is

8! 5 8 ? ? 6 ? ? 4 ? ? 2 ? 1 5

Problem 2

Th e number of permutations of n items of a set arranged r items at a time is

Pn r 5 n!

(n 2 r)! for 0 # r # n.

Example: P10 4 5 10! 6! 5 5040

11. Why can’t r be greater than n?

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Key Concept Number of Permutations

26

10

26 10 10 10 10

7 5 3 40,320

Answers may vary. Sample: n is the number of items you have. r is the

number of items you are selecting. You cannot select more items than

you have.

279_HSM11A2MC_1101_279 279279_HSM11A2MC_1101_279 279 5/18/11 2:17:34 PM5/18/11 2:17:34 PM

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Chapter 11 280

Finding Cn r

Got It? What is the value of C8 3?

15. Cross out the equations that do NOT give the correct formula for C8 3.

C8 3 5 8!

3!(8 2 3)! C8 3 5 8! C8 3 5

8! (8 2 3)!

16. Simplify the remaining equation from Exercise 15.

Problem 4

Th e number of combinations of n items of a set chosen r items at a time is

Cn r 5 n!

r!(n 2 r)! for 0 # r # n

Example: C5 3 5 5!

3!(5 2 3)! 5

5! 3! ? 2! 5

120 6 ? 2 5 10

14. Which is greater, C5 3 or P5 3? Explain.

Key Concept Number of Combinations

Finding Pn r

Got It? In how many ways can 15 runners finish first, second, and third?

12. Use the permutation formula. Circle the value of n, the number of runners in the set. Underline the value of r, the number of runners arranged at a time.

1 2 3 15

13. Use the justifications at the right to find the number of ways in which 15 runners can finish first, second, and third.

Pn r 5 n!

(n 2 r)! Write the formula.

5 !

2 !RQ Substitute n and r.

5 !

! 5 Simplify.

Problem 3

15

15 3

15

12 2730

P5 3 S C5 3 because P5 3 5 n!

(n 2 r)! 5 5!

(5 2 3)! 5 5! 2! 5 60

and C5 3 5 10

C8 3 5 8!

3!(8 2 3)! 5 8!

3! ? 5! 5 8 ? 7 ? 6 ? 5! 3 ? 2 ? 1 ? 5!

5 56

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Lesson Check

281 Lesson 11-1

• Do you UNDERSTAND?

Reasoning Use the definition of permutation to show why 0! should equal 1.

20. Circle the equation that shows the Fundamental Counting Principle and the Permutation Formula for n items arranged n at a time.

0! 5 n ! (n 2 0)!

n! 5 n! (n 2 n)!

n! 5 n ! (n 2 0)!

21. Simplify the equation you chose in 22. Underline the correct expressions to complete Exercise 20. the sentence.

For n!0! to equal 0! / n! , 0! must equal 0 / 1 .

Math Success

Now I get it!

Need to review

0 2 4 6 8 10

Check off the vocabulary words that you understand.

Fundamental Counting Principle permutation n factorial combination

Rate how well you can fi nd permutations and combinations.

Identifying Whether Order Is Important

Got It? A chemistry teacher has a class work in groups to draw the molecular structure of water. Each group submits one drawing. There are eight groups. The teacher selects the four drawings that earn the highest grades. In how many ways can he select and arrange the four drawings from left to right on the wall?

17. Circle the formula you will use to solve this problem.

nCr 5 n!

r!(n 2 r)! nPr 5

n! (n 2 r)!

18. Identify each value.

n 5 r 5

19. In how many different ways can the teacher select and arrange the drawings?

Problem 5

P8 4 5 8!

(8 2 4)! 5 8! 4! 5

8 ? 7 ? 6 ? 5 ? 4! 4! 5 1680

8 4

n! 5 n!(n 2 n)!

n! 5 n!0!

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Vocabulary

Chapter 11 282

11-2 Probability

Review

1. Draw a line from each experiment in Column A to a corresponding set of possible outcomes in Column B.

Column A Column B

Toss a coin. G, R, B

Roll a six-sided number cube. heads or tails

Draw a ball at random from a box holding 1, 2, 3, 4, 5, 6 1 green, 1 red, and 1 blue ball.

Vocabulary Builder

theoretical probability (noun)

thee uh RET ih kul prah buh BIL uh tee

Definition: The theoretical probability of an event is the ratio of the number of ways that the event can occur to the total number of equally likely outcomes in the sample space.

Example: A student rolls a six-sided number cube. The theoretical probability that the student rolls an even number is

P(even) 5 number of ways to roll an even number

number of possible outcomes 5 3 6

Use Your Vocabulary

2. Write T for true or F for false.

Theoretical probability is the sum of the number of ways an event can occur and the number of possible outcomes.

The ratio of the number of ways for an event to occur to the total number of possible outcomes is the theoretical probability.

probability

number of times the event occurs

number of trialsP(event)

F

T

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