Question 1

Use a tree diagram showing all possible results when four fair coins are tossed. Then list the ways of getting the indicated
result.
-more than two tails

Accepted Answer

A) httt, thtt, ttht, ttth 
B) httt, thtt, ttht, ttth, tttt 
C) hhtt, htht, htth, httt, thht, thth, thtt, tthh, ttht, ttth, tttt 
D) hhtt, htht, htth, thht, thth, tthh 
A) httt, thtt, ttht, ttth 
B) httt, thtt, ttht, ttth, tttt 
C) hhtt, htht, htth, httt, thht, thth, thtt, tthh, ttht, ttth, tttt 
D) hhtt, htht, htth, thht, thth, tthh

Question 2

Solve the problem.
-A salesman packed 3 shirts and 6 ties. With one shirt, he could wear all 6 ties. With another shirt, he could wear 5 ties. With the third shirt, he could wear only 3 ties. How many different
Combinations did he have?

Accepted Answer

A) 14 
B) 90 
C) 33 
D) 13 
A) 14 
B) 90 
C) 33 
D) 13

Question 3

Solve the problem.
-How many ways can a president, vice-president, secretary, and treasurer be chosen from a club with 9 members? Assume that no member can hold more than one office.

Accepted Answer

A) 24 
B) 126 
C) 3024 
D) 36 
A) 24 
B) 126 
C) 3024 
D) 36

Question 4

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result.
-The first and second die are even.

Accepted Answer

A) (2,2),(4,4),(6,6) 
B) (2,4),(2,6),(4,2),(4,6)(6,2),(6,4) 
C) (2,2),(2,4),(2,6),(4,2),(4,4),(4,6),(6,2),(6,4),(6,6) 
D) (2,2),(2,4),(2,6) 
A) (2,2),(4,4),(6,6) 
B) (2,4),(2,6),(4,2),(4,6)(6,2),(6,4) 
C) (2,2),(2,4),(2,6),(4,2),(4,4),(4,6),(6,2),(6,4),(6,6) 
D) (2,2),(2,4),(2,6)

Question 5

Read each combination value directly from Pascal's Triangle.
-${ } _ { 6 } C _ { 6 }$

Accepted Answer

A)  720 
B)  $0.5$ 
C)  180 
D)  1 
A)  720 
B)  $0.5$ 
C)  180 
D)  1

Question 6

Determine the number of figures (of any size)in the design.
-Triangles (of any size)

Accepted Answer

A)  13 
B)  16 
C)  19 
D)  15 
A)  13 
B)  16 
C)  19 
D)  15

Question 7

Use a tree diagram showing all possible results when four fair coins are tossed. Then list the ways of getting the indicated
result.
-exactly two tails

Accepted Answer

A) hhtt, htht, htth, thth, tthh 
B) tthh 
C) hhtt, htht, htth, thht, thth, tthh, ttht 
D) hhtt, htht, htth, thht, thth, tthh 
A) hhtt, htht, htth, thth, tthh 
B) tthh 
C) hhtt, htht, htth, thht, thth, tthh, ttht 
D) hhtt, htht, htth, thht, thth, tthh

Question 8

Solve the problem.
-In how many ways can 6 people line up for play tickets?

Accepted Answer

A) 1 
B) 720 
C) 46,656 
D) 6 
A) 1 
B) 720 
C) 46,656 
D) 6

Question 9

Solve the problem.
-A businessman is planning to visit at least 3 but not all of the following cities: Atlanta, Buffalo, Chicago, Detroit, Evanston, Fresno, Gainsville, and Houston. In how many ways can he select the
Set of cities to be visited?

Accepted Answer

A) 232 ways 
B) 162 ways 
C) 219 ways 
D) 218 ways 
A) 232 ways 
B) 162 ways 
C) 219 ways 
D) 218 ways

Question 10

Find the number of ways to get the following card combinations from a 52-card deck.
-Three of one kind and two of another kind (such as three aces and two nines)

Accepted Answer

A) 44,928 ways 
B) 24 ways 
C) 3744 ways 
D) 1872 ways 
A) 44,928 ways 
B) 24 ways 
C) 3744 ways 
D) 1872 ways

Question 11

Solve the problem.
-A signal is made by placing 3 flags, one above the other, on a flag pole. If there are 8 different flags available, how many possible signals can be flown?

Accepted Answer

A) 512 
B) 56 
C) 336 
D) 24 
A) 512 
B) 56 
C) 336 
D) 24

Question 12

Find the number of ways to get the following card combinations from a 52-card deck.
-If two cards are drawn from a 52-card deck without replacement, in how many different ways is it possible to obtain a heart on the first draw and an ace on the second?

Accepted Answer

A) 51 ways 
B) 3 ways 
C) 48 ways 
D) 50 ways 
A) 51 ways 
B) 3 ways 
C) 48 ways 
D) 50 ways

Question 13

Provide an appropriate response.
-Consider the following counting problem. Allison is trying to decide which three of her eight new books to take on vacation with her. How many different ways can she choose the three books?
To solve this problem which of the following rules would you use?

Accepted Answer

A) The combinations rule only 
B) Both the permutations rule and the fundamental counting principle 
C) The permutations rule only 
D) The fundamental counting principle only 
A) The combinations rule only 
B) Both the permutations rule and the fundamental counting principle 
C) The permutations rule only 
D) The fundamental counting principle only

Question 14

Solve the problem.
-If you toss four fair coins, in how many ways can you obtain at least one head?

Accepted Answer

A) 16 ways 
B) 4 ways 
C) 15 ways 
D) 5 ways 
A) 16 ways 
B) 4 ways 
C) 15 ways 
D) 5 ways

Question 15

Evaluate the factorial expression.
-$\frac { \mathrm { n } ! } { \mathrm { r } ! ( \mathrm { n } - \mathrm { r } ) ! }$, where $\mathrm { n } = 38$ and $\mathrm { r } = 9$

Accepted Answer

A)  50,616 
B)  1406 
C)  $1,771,560$ 
D)  $163,011,640$ 
A)  50,616 
B)  1406 
C)  $1,771,560$ 
D)  $163,011,640$

Question 16

Find the number of ways to get the following card combinations from a 52-card deck.
-Two red cards and three black cards

Accepted Answer

A) 422,500 ways 
B) 1,690,000 ways 
C) 1,267,500 ways 
D) 845,000 ways 
A) 422,500 ways 
B) 1,690,000 ways 
C) 1,267,500 ways 
D) 845,000 ways

Question 17

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result.
-Exactly one die shows a 3.

Accepted Answer

A) (3,3) 
B) (3, 1),(3, 2),(3, 3), (3, 4),(3, 5),(3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3) 
C) (3, 1),(3, 2),(3, 4),(3, 5),(3, 6) 
D) (3, 1),(3, 2),(3, 4),(3, 5),(3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3) 
A) (3,3) 
B) (3, 1),(3, 2),(3, 3), (3, 4),(3, 5),(3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3) 
C) (3, 1),(3, 2),(3, 4),(3, 5),(3, 6) 
D) (3, 1),(3, 2),(3, 4),(3, 5),(3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3)

Question 18

Evaluate the permutation.
-${ } _ { 6 } \mathrm { P } _ { 2 }$

Accepted Answer

A)  6 
B)  1 
C)  120 
D)  30 
A)  6 
B)  1 
C)  120 
D)  30

Question 19

Find the number of ways to get the following card combinations from a 52-card deck.
-If two cards are drawn from a 52-card deck without replacement, in how many different ways is it possible to obtain a king on the first draw and a king on the second?

Accepted Answer

A) 2 ways 
B) 8 ways 
C) 16 ways 
D) 12 ways 
A) 2 ways 
B) 8 ways 
C) 16 ways 
D) 12 ways

Question 20

Evaluate the permutation.
-Determine the number of permutations of 10 things taken 7 at a time.

Accepted Answer

A) 720 
B) 1 
C) 604,800 
D) 10 
A) 720 
B) 1 
C) 604,800 
D) 10

Use a tree diagram showing all possible results when four fair coins are tossed. Then list the ways of getting the indicated result. -more than two tails

Solve the problem. -A salesman packed 3 shirts and 6 ties. With one shirt, he could wear all 6 ties. With another shirt, he could wear 5 ties. With the third shirt, he could wear only 3 ties. How many different Combinations did he have?

Solve the problem. -How many ways can a president, vice-president, secretary, and treasurer be chosen from a club with 9 members? Assume that no member can hold more than one office.

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result. -The first and second die are even.

Read each combination value directly from Pascal's Triangle. - ${ } _ { 6 } C _ { 6 }$

Determine the number of figures (of any size)in the design. -Triangles (of any size)

Use a tree diagram showing all possible results when four fair coins are tossed. Then list the ways of getting the indicated result. -exactly two tails

Solve the problem. -In how many ways can 6 people line up for play tickets?

Solve the problem. -A businessman is planning to visit at least 3 but not all of the following cities: Atlanta, Buffalo, Chicago, Detroit, Evanston, Fresno, Gainsville, and Houston. In how many ways can he select the Set of cities to be visited?

Find the number of ways to get the following card combinations from a 52-card deck. -Three of one kind and two of another kind (such as three aces and two nines)

Solve the problem. -A signal is made by placing 3 flags, one above the other, on a flag pole. If there are 8 different flags available, how many possible signals can be flown?

Find the number of ways to get the following card combinations from a 52-card deck. -If two cards are drawn from a 52-card deck without replacement, in how many different ways is it possible to obtain a heart on the first draw and an ace on the second?

Provide an appropriate response. -Consider the following counting problem. Allison is trying to decide which three of her eight new books to take on vacation with her. How many different ways can she choose the three books? To solve this problem which of the following rules would you use?

Solve the problem. -If you toss four fair coins, in how many ways can you obtain at least one head?

Evaluate the factorial expression. - $\frac { \mathrm { n } ! } { \mathrm { r } ! ( \mathrm { n } - \mathrm { r } ) ! }$ , where $\mathrm { n } = 38$ and $\mathrm { r } = 9$

Find the number of ways to get the following card combinations from a 52-card deck. -Two red cards and three black cards

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result. -Exactly one die shows a 3.

Evaluate the permutation. - ${ } _ { 6 } \mathrm { P } _ { 2 }$

Find the number of ways to get the following card combinations from a 52-card deck. -If two cards are drawn from a 52-card deck without replacement, in how many different ways is it possible to obtain a king on the first draw and a king on the second?

Evaluate the permutation. -Determine the number of permutations of 10 things taken 7 at a time.

The Art of Problem Solving

The Basic Concepts of Set Theory

Introduction to Logic

Numeration Systems

Number Theory

The Real Numbers and Their Representations

The Basic Concepts of Algebra

Graphs, Functions, and Systems of Equations and Inequalities

Geometry

Probability

Statistics

Personal Financial Management

Trigonometry Formerly

Graph Theory

Voting and Apportionment

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Exam 10: Counting Methods