homework, page 786

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 1

Homework, Page 786

Evaluate the expression by hand, then check your work with a calculator.

121.

5

12 12! 12! 12 11 10 9 8 7!

5 5! 12 5 ! 5!7! 5 4 3 2 1 7!

12 11 10 9 8 11 9 8792

5 4 3 2 1 1


Homework, Page 786

Evaluate the expression by hand, then check your work with a calculator.

12 75. P

12 7

12! 12! 12 11 10 9 8 7 6 5!12 11 10 9 8 7 6

12 7 ! 5! 5!

132 720 42 95040 42 3991680

P


Homework, Page 786

9. How many license plates begin with two letters followed by four digits or begin with three digits followed by three letters. Assume that no digits or letters are repeated.

Two letters followed by four digits:

26 25 10 9 8 7 3,276,000

Three digits followed by three letters:

10 9 8 26 25 24 11,232,000

3,276,000 11,232,000 14,508,000


Homework, Page 786

13. Suppose that a coin is tossed five times. How many different outcomes include at least two heads?

5If a coin is tossed five times, there are 2 32 possible outcomes.

One outcome is all tails, five are four tails and one head, all others

have two or more heads, so 32 1 5 26 possible outcomes have

at l

east two heads.


Homework, Page 786

17. Find the number of distinguishable permutations that can be made from the letters in:

A) GERMANY

B) PRESBYTERIANSThe thirteen letters may be made into

13!778,377,600 distinguishable permutaitons.

2! 2! 2!

The seven letters may be made into

7! 5040 distinguishable permutations.


Homework, Page 786

Expand each expression.

21. 52 33x y

5 5 4 1 3 2 2 32 3 2 2 3 2 3 2 3

4 52 3 3

10 8 3 6 6 4 9

2 12 15

From Pascal's triangle,

3 3 5 3 10 3 10 3

5 3

243 5 81 10 27 10 9

15

243

x y x x y x y x y

x y y

x x y x y x y

x y y

10 8 3 6 6 4 9 2 12 15405 270 90 15x x y x y x y x y y


Homework, 86

25. 118Find the coefficient of in the expansion of 2x x

38 8 8

8

8 3 11 1 3

112 165 8 1320

3

The coefficient of is 1320

i

x x x

x


Homework, Page 786

List the elements of the sample space.

29. A two digit code is selected from the digits 1,3,6 where no digits

are repeated.

3 2 6 1,3 , 1,6 , 3,1 , 6,1 , 3,6 , 6,3P


Homework, Page 786

A penny, a nickel, and a dime are tossed.

33. List all the outcomes in the complement of the event "two heads

or two tails."

2 2 2 8 T,T,T , H,H,H

The other six outcomes have either two heads or two tails.


Homework, Page 786

37. A fair coin is tossed four times. Find the probability of obtaining one head and three tails.

2 2 2 2 16 H,T,T,T , T,H,T,T , T,T,H,T , T,T,T,H

4 1The probability of one head and three tails is or .16 4


Homework, Page 786

An experiment has only two possible outcomes – success (S) or failure (F) – and repetitions are independent events. Probability of success is 0.4.

41. Find the probability of SF on two repetitions.

P SF 0.4 0.6 0.24


Homework, Page 78645. Two cans of mixed nuts of different brands are open on a table. Brand A consists of 30% cashews, while brand B consists of 40% cashews. A can is chosen at random and a nut is chosen at random from the can. Find the probability that the nut is:

a) from the brand A can.

b) a brand A cashew

c) a cashew

d) from the brand A can, given that it is a cashew

P A 0.5

P A and cashew 0.5 0.3 0.15

P cashew 0.5 0.3 0.5 0.4 0.35

P A and cashew 0.15P A|cashew 0.429

P cashew 0.35


Homework, Page 786

Find the first six terms and the 12th term of the sequence.

49. 1 11 and 3, for 2n na a a n

1

2 1

3

4

5

6

12 1

1

2 2 1 3 3 3 1

5

8

11

14

3 12 1 1 33 32

n

a

a d a a n

a

a

a

a

a a


Homework, Page 786

Find the first six terms and the 12th term of the sequence.

53. 1 2 2 13, 1, and , for 3k k kv v v v v k

1 2 3 4

5 6 7

8 9 10

11 12

3, 1, 3 1 2, 1 2 1

2 1 3, 1 3 4, 3 4 7

4 7 11, 7 11 18, 11 18 29

18 29 47, 29 47 76

v v v v

v v v

v v v

v v


Homework, Page 786

The sequences are arithmetic or geometric. Find an explicit formula for the nth term. State the common difference or ratio.

57. 10,12,14.4,17.28,

1

10,12,14.4,17.28,

12 10 2 12 2 14.4

121.2 12 1.2 14.4 Common ratio 1.2

10

10 1.2nna


Homework, Page 786

The sequences are arithmetic or geometric. Find an explicit formula for the nth term. State the common difference or ratio.

61. The fourth and ninth terms of a geometric sequence are –192 and 196,608, respectively.

5 5

353

1

_, _, _, 192, _, _, _, _,196608

196608196608 192 1024

192192

1024 4 192 4 34

3 4 ; 4n

n

r r

a a

a r


Homework, Page 786

Find the sum of the terms of the arithmetic sequence.

65. 2.5, 0.5, 3.5, , 75.5

27

1

2.5, 0.5, 3.5, , 75.5

0.5 2.5 3 3

75.5 2.5 3 1 78 3 3 3 81 27

2.5 75.52.5 3 1 27 985.5

2n

d

n n n n

n


Homework, Page 786

Find the sum of the geometric sequence.

69. 2,6,18, ,39366

1 1

9

10101

1

2,6,18, ,39366

6 393663 3 39366 2 3 3 19683

2 2

3 19683 1 9 10

1 32 3 2 59,048

1 3

n n

n

n

r

n n


Homework, Page 786Graph the sequence.

73. 11

n

na n


Homework, Page 786Determine whether the geometric series converges. If it does, find its sum.

77. 32

4

j

j i

32 converges

4

j

j i

3 because 1

4

1

32

4a

1.5

132

4 1

j

j i

a

r

1.5

31

4

1.5

0.25 6


Homework, Page 786Determine whether the geometric series converges. If it does, find its sum.

81. 3 0.5k

k i

3 0.5 converges k

k i

because 0.5 1

1 3 0.5a 1.5

13 0.51

k

k i

a

r

1.5

1 0.5

1.5

0.5 3


Homework, Page 786Write the sum in sigma notation.

85. 2 2 21 3 5

22 1na n

2

12 1

nn


Homework, Page 786Use summation formulas to evaluate the expression.

89. 252

13 4

kk k

2

1

1 2 1

6

n

k

n n nk

from Example 2, page 754

1

3 13

2

n

k

n nk

from exercise 23, page 756

1

4 4 n

kn

from the definition of multiplication

252

13 4

kk k

25 25 1 2 25 1 3 25 25 1

4 256 2

25 26 51 3 25 26100

6 2

5525 975 100 4650


Homework, Page 786Use mathematical induction to prove the statement is true for all positive integers.

93.12 !n n

1 11 : 2 1!P 0 1

1: 2 !kkP k

1 1 11 : 2 2 2k k

kP

2 !k 1 !k k 1 !k 12 ! for all 1n n n


Homework, Page 786Construct a) a stemplot, b) a frequency table, and c) a histogram for the indicated data.

97. Use intervals of ten. The lengths (in seconds) of 24 randomly selected Beatles songs that appeared on singles are as follows, in the order released: 143, 120, 120, 139, 124, 144, 131, 132, 148, 163, 140, 177, 136, 124, 179, 131, 180, 137, 156, 202, 191, 197, 230, 190.


Homework, Page 78697. a) a stemplot

12 0 0 4 4

13 9 1 2 6 2 7

14 3 4 8 0

15 6

16 3

17 7 9

18 0

19 1 7 0

20 2

21

22

23 0

Seconds


Homework, Page 78697. b) a frequency table:

Seconds Frequency Seconds Frequency

120-129 4 180-189 1

130-139 6 190-199 3

140-149 4 200-209 1

150-159 1 210-219 0

160-169 1 220-229 0

170-179 2 230-239 1

Total 24


Homework, Page 78697. c) a histogram

Histogram

0

2

4

6

8

120

140

160

180

200

220

240

Time (Seconds)

Fre

qu

ency

Frequency


Homework, Page 786Find the five number summary, the range, the interquartile range, the standard deviation, and the variance of the specified data. Identify any outliers.

101. The data in Exercise 95.

Five number summary: {9.1, 11.7, 13.1, 15.4, 23.4}

Range: 14.3

IQR: 4.7

Standard deviation: 3.185

Variance:10.4454.7 1.5 7.05

11.7 7.05 4.65 9.1

15.4 7.05 22.45 23.4 is an outlier


Homework, Page 786Construct (a) a boxplot and (b) a modified boxplot for the specified data.

105. The data in Exercise 97.

Five number summary: {120, 131.5, 143.5, 179.5, 2304}

IQR: 48

(a) (b)

48 1.5 64

131.5 64 67.5 120

179.5 64 243.5 230, no outliers


Homework, Page 786109. Make a time plot for the data in Exercise 97 assuming equal time between songs. Interpret the trend revealed in the time plot.

The time plot reveals a trend toward longer songs over time.


Homework, Page 786113. Suppose the probability of producing a defective baseball bat is 0.02. Four bats are selected at random. What is the probability that the lot of four bats contains the following?

a) No defective bats

b) One defective bat.

40 0.98 0.922P

341 0.98 0.02 0.0188

1P

homework, page 786

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