College Algebra 9th txtbk

Review Exercises 567
33. From a standard deck of 52 cards, what is the probability of obtaining
a 5-card hand:
(A) Of all diamonds?
(B) Of three diamonds and two spades?
Write answers in terms of C n,r or P n,r , as appropriate. Do not
evaluate.
34. A group of 10 people includes one married couple. If four people
are selected at random, what is the probability that the married
couple is selected?
35. A spinning device has three numbers, 1, 2, 3, each as likely to
turn up as the other. If the device is spun twice, what is the probability
that:
(A) The same number turns up both times?
(B) The sum of the numbers turning up is 5?
36. Use the formula for the sum of an infinite geometric series to
write 0.727 272 . . . 0.72 as the quotient of two integers.
37. Solve the following problems using P n,r or C n,r , as appropriate:
(A) How many three-digit opening combinations are possible
on a combination lock with six digits if the digits cannot
be repeated?
(B) Suppose five tennis players have made the finals. If each
of the five players is to play every other player exactly
once, how many games must be scheduled?
Evaluate Problems 38–40.
20!
38. 39. a16 40.
18!(20 18)!
12 b
41. Expand (x y) 5 using the binomial formula.
42. Find the term containing x 6 in the expansion of (x 2) 9 .
43. If the terms in the expansion of (2x y) 12 are arranged in descending
powers of x, find the tenth term.
Establish each statement in Problems 44–46 for all natural
numbers using mathematical induction.
44. P n in Problem 17 45. P n in Problem 18
46. P n in Problem 19
In Problems 47 and 48, find the smallest positive integer n such
that a n b n by graphing the sequences {a n } and {b n } with a
graphing calculator. Check your answer by using a graphing
calculator to display both sequences in table form.
47. a n C 50,n , b n 3 n
48. a 1 100, a n 0.99a n1 5, b n 9 7n
a 11
11 b
49. How many different families with five children are possible, excluding
multiple births, where the sex of each child in the order of
their birth is taken into consideration? How many families are
possible if the order pattern is not taken into account?
50. A free-falling body travels g/2 feet in the first second, 3g2 feet
during the next second, 5g2 feet the next, and so on. Find the
distance fallen during the twenty-fifth second and the total distance
fallen from the start to the end of the twenty-fifth second.
51. How many ways can two people be seated in a row of four chairs?
52. Expand (x i) 6 , where i is the imaginary unit, using the binomial
formula.
53. If three people are selected from a group of seven men and
three women, what is the probability that at least one woman is
selected?
54. Three fair coins are tossed 1,000 times with the following frequencies
of outcomes:
Number of heads 0 1 2 3
Frequency 120 360 350 170
(A) What is the approximate empirical probability of obtaining
two heads?
(B) What is the theoretical probability of obtaining two heads?
(C) What is the expected frequency of obtaining two heads?
Prove that each statement in Problems 55–59 holds for all positive
integers using mathematical induction.
55.
n
a
k 1
56. x 2n y 2n is divisible by x y, x y
a n
k 3 a a
n
k 1
2
kb
57. n 7 m; n, m positive integers
a m anm ;
58. {a n } {b n }, where a n a n1 2, a 1 3, b n 5 2n
59. (1!)1 (2!)2 (3!)3 ... (n!)n (n 1)! 1(From
U.S.S.R. Mathematical Olympiads, 1955–1956, Grade 10.)
APPLICATIONS
60. LOAN REPAYMENT You borrow $7,200 and agree to pay 1% of
the unpaid balance each month for interest. If you decide to pay an additional
$300 each month to reduce the unpaid balance, how much interest
will you pay over the 24 months it will take to repay this loan?
61. ECONOMICS Due to reduced taxes, an individual has an extra
$2,400 in spendable income. If we assume that the individual spends
75% of this on consumer goods, and the producers of those consumer
goods in turn spend 75% on consumer goods, and that this process
continues indefinitely, what is the total amount (to the nearest dollar)
spent on consumer goods?
62. COMPOUND INTEREST If $500 is invested at 6% compounded
annually, the amount A present after n years forms a geometric sequence
with common ratio 1 0.06 1.06. Use a geometric sequence
formula to find the amount A in the account (to the nearest
cent) after 10 years; after 20 years.
63. TRANSPORTATION A distribution center A wishes to distribute
its products to five different retail stores, B, C, D, E, and F, in a city.
How many different route plans can be constructed so that a single
truck can start from A, deliver to each store exactly once, and then return
to the center?
64. MARKET ANALYSIS A DVD distributor selected 1,000 persons
at random and surveyed them to determine a relationship between
age of purchaser and annual DVD purchases. The results are given in
the table on page 568.