Homework, Page 786
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Transcript of Homework, Page 786
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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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 2
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 3
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 4
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.
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 5
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.
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 6
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 7
Homework, Page 786
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 8
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 9
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.
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 10
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 11
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 12
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 13
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 14
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 15
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 16
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 17
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 18
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 19
Homework, Page 786Graph the sequence.
73. 11
n
na n
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 20
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 21
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 22
Homework, Page 786Write the sum in sigma notation.
85. 2 2 21 3 5
22 1na n
2
12 1
nn
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 23
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 24
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 25
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.
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 26
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 27
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 28
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 29
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 30
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
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 31
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.
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 32
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