Question 1

In how many different ways can a test consisting of seven true-or-false questions be completed?
A)49
B)128
C)256
D)14
E)9

Accepted Answer

Each question has 2 possible answers (true or false), and there are 7 questions. Thus, the number of different ways to complete the test is 2^7 = 128.

Question 2

Without expanding completely, find the middle term in the expansion of the expression. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Without expanding completely, find the middle term...

Question 3

A girl has three skirts, five blouses and four sweaters. How many different skirt-blouse-sweater combinations can she wear?
A)70
B)60
C)12
D)56
E)45

Accepted Answer

The number of combinations is calculated by multiplying the number of choices for each item: 3 skirts * 5 blouses * 4 sweaters = 60 combinations.

Question 4

Find the number. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the number. 
A) 
B) 
C) 
D)...

Question 5

Use the principle of mathematical induction to find the equivalent expression for every positive integer n. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use the principle of mathematical induction to...

Question 6

Find the total length of the red-line curve in the figure if the width of the maze formed by the curve is 32 inches and all halls in the maze have width 2 inches.   in. and  in.
A)548 inches
B)544 inches
C)538 inches
D)540 inches
E)612 inches

Accepted Answer

The answer of Find the total length of the red-line...

Question 7

A contest will have five cash prizes totaling $6,000, and there will be a $100 difference between successive prizes. Find the first prize.
A)$1,900
B)$1,400
C)$1,700
D)$1,600
E)$1,800

Accepted Answer

The answer of A contest will have five cash prizes...

Question 8

Find the nth term and the fifth term of the geometric sequence. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the nth term and the fifth...

Question 9

A man is 54 years old and a woman is 34 years old. The probability that the man will be alive in 10 years is 0.77, whereas the probability that the woman will be alive 10 years from now is 0.94. Assume that their life expectancies are unrelated. 
A. Find the probability that they will both be alive 10 years from now.
B. Find the probability that neither one will be alive 10 years from now.
C. Find the probability that at least one of the two will be alive 10 years from now.

A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of A man is 54 years old and...

Question 10

If eleven basketball teams are in a tournament, find the number of different ways that first, second, and third place can be decided, assuming ties are not allowed.
A)110
B)6
C)7,920
D)11,880
E)990

Accepted Answer

The number of different ways to decide first, second, and third place among eleven teams, assuming no ties, is calculated using permutations since the order matters. The formula for permutations is $nPr = \frac{n!}{(n-r)!}$, where $n$ is the total number of items, and $r$ is the number of items to choose. Here, $n = 11$ (teams) and $r = 3$ (places). Thus, the calculation is $11P3 = \frac{11!}{(11-3)!} = \frac{11!}{8!} = 11 \times 10 \times 9 = 990$.

Question 11

During 1990, smoking caused 418,890 deaths in the United States. Of these deaths, cardiovascular disease accounted for 179,820, cancer for 151,322, and respiratory diseases such as emphysema for 84,475. Find the probability that a smoking-related death was the result of either cardiovascular disease or cancer.
A)0.791
B)0.803
C)0.786
D)0.813
E)0.209

Accepted Answer

The probability is found by adding the deaths from cardiovascular disease and cancer, then dividing by the total deaths: (179,820 + 151,322) / 418,890 = 331,142 / 418,890 = 0.791.

Question 12

Express the sum in terms of n.  (Hint: Use the theorem on sums to write the sum as  . Next, use the following formulas  and the theorem on the sum of a constant.)
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Express the sum in terms of n....

Question 13

To win a state lottery game, a player must correctly select six numbers from the numbers 1 through 49. Find the total number of selections possible if a player selects only even numbers.
A)144
B)96,909,120
C)6,991,908
D)134,596
E)60,410,085,120

Accepted Answer

There are 24 even numbers between 1 and 49 (inclusive). The total number of ways to select 6 numbers out of these 24 is calculated using combinations (since the order of selection does not matter), which is given by the formula $C(n, r) = \frac{n!}{r!(n-r)!}$, where $n$ is the total number of items to choose from, $r$ is the number of items to choose, and $n!$ denotes the factorial of $n$. Therefore, the total number of selections possible is $C(24, 6) = \frac{24!}{6!(24-6)!} = \frac{24!}{6!18!} = 134,596$.

Question 14

A bicycle rider coasts downhill, traveling 7 feet the first second. In each succeeding second, the rider travels 4 feet farther than in the preceding second. If the rider reaches the bottom of the hill in 8 seconds, find the total distance traveled.
A)164 feet
B)172 feet
C)160 feet
D)152 feet
E)168 feet

Accepted Answer

This is an arithmetic sequence problem where the distance traveled each second forms an arithmetic sequence with the first term $a = 7$ feet and the common difference $d = 4$ feet. The total distance traveled in $n = 8$ seconds can be found using the formula for the sum of the first $n$ terms of an arithmetic sequence: $S_n = \frac{n}{2}[2a + (n-1)d]$. Substituting the given values, we get $S_8 = \frac{8}{2}[2(7) + (8-1)4] = 4[14 + 28] = 4[42] = 168$ feet.

Question 15

Use the principle of mathematical induction to find the equivalent expression for every positive integer n. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use the principle of mathematical induction to...

Question 16

Evaluate the expression. 
A)9
B)8
C)90
D)72
E)362,881

Accepted Answer

The answer of Evaluate the expression. 
A)9
B)8
C)90
D)72
E)362,881...

Question 17

A pile of logs has 34 logs in the bottom layer, 32 in the second layer, 30 in the third, and so on. The top layer contains 12 logs. Find the total number of logs in the pile.
A)276 logs
B)270 logs
C)552 logs
D)274 logs
E)272 logs

Accepted Answer

The number of logs forms an arithmetic sequence with the first term 34 and the last term 12, with a common difference of -2. The number of terms can be found using the formula for the nth term of an arithmetic sequence: 12 = 34 + (n-1)(-2). Solving gives n = 12. The sum of the sequence is given by the formula S = n/2 * (first term + last term) = 12/2 * (34 + 12) = 276 logs.

Question 18

A committee of 4 men and 3 women is to be chosen from a group of 11 men and 5 women. Determine the number of different ways of selecting the committee.
A)4
B)330
C)3,300
D)79,200

Accepted Answer

The number of ways to choose 4 men out of 11 is $C(11,4) = \frac{11!}{4!(11-4)!}$, and the number of ways to choose 3 women out of 5 is $C(5,3) = \frac{5!}{3!(5-3)!}$. The total number of ways to form the committee is the product of these two, which is $C(11,4) \times C(5,3) = 330 \times 10 = 3,300$.

Question 19

Find the smallest positive integer j for which the statement is true. 
A)19
B)256
C)16
D)12
E)15

Accepted Answer

The answer of Find the smallest positive integer j for...

Question 20

A committee is going to select 28 students from a pool of 800 to receive scholarships. How may ways could the students be selected if each scholarship is worth the same amount?
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of A committee is going to select 28...

Quiz 10: Sequences, Series and Probability

In how many different ways can a test consisting of seven true-or-false questions be completed?

Without expanding completely, find the middle term in the expansion of the expression.

A girl has three skirts, five blouses and four sweaters. How many different skirt-blouse-sweater combinations can she wear?

Find the number.

Use the principle of mathematical induction to find the equivalent expression for every positive integer n.

Find the total length of the red-line curve in the figure if the width of the maze formed by the curve is 32 inches and all halls in the maze have width 2 inches. in. and in.

A contest will have five cash prizes totaling $6,000, and there will be a $100 difference between successive prizes. Find the first prize.

Find the nth term and the fifth term of the geometric sequence.

If eleven basketball teams are in a tournament, find the number of different ways that first, second, and third place can be decided, assuming ties are not allowed.

Express the sum in terms of n. (Hint: Use the theorem on sums to write the sum as . Next, use the following formulas and the theorem on the sum of a constant.)

To win a state lottery game, a player must correctly select six numbers from the numbers 1 through 49. Find the total number of selections possible if a player selects only even numbers.

A bicycle rider coasts downhill, traveling 7 feet the first second. In each succeeding second, the rider travels 4 feet farther than in the preceding second. If the rider reaches the bottom of the hill in 8 seconds, find the total distance traveled.

Use the principle of mathematical induction to find the equivalent expression for every positive integer n.

Evaluate the expression.

A pile of logs has 34 logs in the bottom layer, 32 in the second layer, 30 in the third, and so on. The top layer contains 12 logs. Find the total number of logs in the pile.

A committee of 4 men and 3 women is to be chosen from a group of 11 men and 5 women. Determine the number of different ways of selecting the committee.

Find the smallest positive integer j for which the statement is true.

A committee is going to select 28 students from a pool of 800 to receive scholarships. How may ways could the students be selected if each scholarship is worth the same amount?