Question 1

Read each combination value directly from Pascal's Triangle.
-${ } _ { 9 } \mathrm { C } _ { 8 }$

Accepted Answer

A)  362,880 
B)  8 
C)  72 
D)  9 
A)  362,880 
B)  8 
C)  72 
D)  9 D

Question 2

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result.
-The sum of the numbers showing is 5.

Accepted Answer

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

Question 3

Using the 36 possibilities found in the product table for rolling two dice, list and count the outcomes for which the sum (for both dice)is the following.
-Less than 4

Accepted Answer

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

Question 4

Solve the problem.
-A saleswoman packed 3 jackets and 6 skirts. With one jacket, she could wear all 6 skirts. With another jacket, she could wear 5 skirts. With the third jacket, she could wear only 2 skirts. How
Many different combinations did she have?

Accepted Answer

A) 13 
B) 12 
C) 32 
D) 60 
A) 13 
B) 12 
C) 32 
D) 60

Question 5

Find the number of ways to get the following card combinations from a 52-card deck.
-All tens in a five-card hand

Accepted Answer

A) 144 ways 
B) 48 ways 
C) 192 ways 
D) 0 ways 
A) 144 ways 
B) 48 ways 
C) 192 ways 
D) 0 ways

Question 6

Given a magic square, other magic squares may be obtained by rotating, adding, or subtracting a constant value to or from each entry, multiplying each entry by a constant, or dividing each entry by a nonzero constant. Start with the given magic square and perform the indicated operation to find a new magic square.
-Add 2
 $\begin{array}{|l|l|l|}
\hline 2 & 9 & 4 \
\hline 7 & 5 & 3 \
\hline 6 & 1 & 8 \
\hline
\end{array}$

Accepted Answer

A) 
$\begin{array}{|c|c|c|}
\hline 4 & 11 & 4 \
\hline 9 & 5 & 5 \
\hline 6 & 3 & 10 \
\hline
\end{array}$ 
B) 
$\begin{array}{|c|c|c|}
\hline 6 & 11 & 4 \
\hline 9 & 7 & 5 \
\hline 10 & 3 & 8 \
\hline
\end{array}$ 
C) 
$\begin{array}{|c|c|c|}
\hline 2 & 11 & 6 \
\hline 9 & 5 & 5 \
\hline 8 & 3 & 8 \
\hline
\end{array}$ 
D) 
$\begin{array}{|c|c|c|}
\hline 4 & 11 & 6 \
\hline 9 & 7 & 5 \
\hline 8 & 3 & 10 \
\hline
\end{array}$ 
A) 
$\begin{array}{|c|c|c|}
\hline 4 & 11 & 4 \
\hline 9 & 5 & 5 \
\hline 6 & 3 & 10 \
\hline
\end{array}$ 
B) 
$\begin{array}{|c|c|c|}
\hline 6 & 11 & 4 \
\hline 9 & 7 & 5 \
\hline 10 & 3 & 8 \
\hline
\end{array}$ 
C) 
$\begin{array}{|c|c|c|}
\hline 2 & 11 & 6 \
\hline 9 & 5 & 5 \
\hline 8 & 3 & 8 \
\hline
\end{array}$ 
D) 
$\begin{array}{|c|c|c|}
\hline 4 & 11 & 6 \
\hline 9 & 7 & 5 \
\hline 8 & 3 & 10 \
\hline
\end{array}$

Question 7

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

Accepted Answer

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

Question 8

Solve the problem.
-How many ways can a president, vice-president, and secretary be chosen from a club with 12 members?

Accepted Answer

A) 1320 
B) 6 
C) 220 
D) 36 
A) 1320 
B) 6 
C) 220 
D) 36

Question 9

How many 10-digit telephone numbers (area code + number)are possible if the first digit cannot be zero, the first three digits cannot be 800 or 900, and the number must end in 0000?

Accepted Answer

A) 900,000 
B) 898,000 
C) 654,642 
D) 899,000 
A) 900,000 
B) 898,000 
C) 654,642 
D) 899,000

Question 10

Solve the problem.
-Maria, Daniel, Stephanie, Michael, Elena, Tyler, Sue, and Dimitri have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if Maria
And Daniel are to sit next to each other?

Accepted Answer

A) 1440 
B) 5040 
C) 11,520 
D) 10,080 
A) 1440 
B) 5040 
C) 11,520 
D) 10,080

Question 11

Of the 2,598,960 different five-card hands possible from a deck of 52 playing cards, how many would contain all black cards.

Accepted Answer

A) 131,560 hands 
B) 263,120 hands 
C) 32,890 hands 
D) 65,780 hands 
A) 131,560 hands 
B) 263,120 hands 
C) 32,890 hands 
D) 65,780 hands

Question 12

A pool of possible jurors consists of 15 men and 10 women. How many different juries consisting of 5 men and 7 women are possible?

Accepted Answer

A) 3123 
B) 1,352,078 
C) 5,200,300 
D) 360,360 
A) 3123 
B) 1,352,078 
C) 5,200,300 
D) 360,360

Question 13

Given a magic square, other magic squares may be obtained by rotating, adding, or subtracting a constant value to or from each entry, multiplying each entry by a constant, or dividing each entry by a nonzero constant. Start with the given magic square and perform the indicated operation to find a new magic square.
-Divide by 4
 $\begin{array}{|c|c|c|}
\hline 32 & 4 & 24 \
\hline 12 & 20 & 28 \
\hline 16 & 36 & 8 \
\hline
\end{array}$

Accepted Answer

A) 
$\begin{array}{|l|l|l|}
\hline 8 & 1 & 6 \
\hline 12 & 20 & 28 \
\hline 4 & 9 & 2 \
\hline
\end{array}$ 
B) 
$\begin{array}{|l|l|l|}
\hline 8 & 1 & 6 \
\hline 3 & 5 & 7 \
\hline 4 & 9 & 2 \
\hline
\end{array}$ 
C) 
$\begin{array}{|c|c|c|}
\hline 32 & 4 & 24 \
\hline 3 & 5 & 7 \
\hline 4 & 9 & 2 \
\hline
\end{array}$ 
D) 
$\begin{array}{|c|c|c|}
\hline 8 & 1 & 6 \
\hline 3 & 5 & 7 \
\hline 16 & 36 & 8 \
\hline
\end{array}$ 
A) 
$\begin{array}{|l|l|l|}
\hline 8 & 1 & 6 \
\hline 12 & 20 & 28 \
\hline 4 & 9 & 2 \
\hline
\end{array}$ 
B) 
$\begin{array}{|l|l|l|}
\hline 8 & 1 & 6 \
\hline 3 & 5 & 7 \
\hline 4 & 9 & 2 \
\hline
\end{array}$ 
C) 
$\begin{array}{|c|c|c|}
\hline 32 & 4 & 24 \
\hline 3 & 5 & 7 \
\hline 4 & 9 & 2 \
\hline
\end{array}$ 
D) 
$\begin{array}{|c|c|c|}
\hline 8 & 1 & 6 \
\hline 3 & 5 & 7 \
\hline 16 & 36 & 8 \
\hline
\end{array}$

Question 14

Solve the problem.
-A restaurant offered salads with 3 type(s)of dressing and 6 different topping(s). How many different types of salad could be ordered?

Accepted Answer

A) 36 
B) 18 
C) 9 
D) 9. 
A) 36 
B) 18 
C) 9 
D) 9.

Question 15

If two fair dice, one red and one white, are rolled, in how many ways can the result be obtained?
-A different number on each die.

Accepted Answer

A) 6 ways 
B) 36 ways 
C) 30 ways 
D) 25 ways 
A) 6 ways 
B) 36 ways 
C) 30 ways 
D) 25 ways

Question 16

Solve the problem.
-A baseball manager has 10 players of the same ability. How many 9 player starting lineups can he create?

Accepted Answer

A) 10 
B) 362,880 
C) 90 
D) 3,628,800 
A) 10 
B) 362,880 
C) 90 
D) 3,628,800

Question 17

Solve the problem.
-How many 5-card poker hands consisting of 2 aces and 3 kings are possible with an ordinary 52-card deck?

Accepted Answer

A) 24 
B) 12 
C) 6 
D) 288 
A) 24 
B) 12 
C) 6 
D) 288

Question 18

Solve the problem.
-A chef makes 3 kinds of fruit salad. Raisins, coconut, almonds, or marshmallows can be added to each kind of salad. How many different fruit salads are possible?

Accepted Answer

A) 64 salads 
B) 24 salads 
C) 32 salads 
D) 48 salads 
A) 64 salads 
B) 24 salads 
C) 32 salads 
D) 48 salads

Question 19

The library is to be given 10 books as a gift. The books will be selected from a list of 23 titles. If each book selected must have a different title, how many possible selections are there?

Accepted Answer

A)  $1,144,066$ 
B)  $4.151586701 \mathrm { e } + 12$ 
C)  $7.124122779 \mathrm { e } + 15$ 
D)  $2.585201674 \mathrm { e } + 22$ 
A)  $1,144,066$ 
B)  $4.151586701 \mathrm { e } + 12$ 
C)  $7.124122779 \mathrm { e } + 15$ 
D)  $2.585201674 \mathrm { e } + 22$

Question 20

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

Accepted Answer

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

Read each combination value directly from Pascal's Triangle. - ${ } _ { 9 } \mathrm { C } _ { 8 }$

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result. -The sum of the numbers showing is 5.

Using the 36 possibilities found in the product table for rolling two dice, list and count the outcomes for which the sum (for both dice)is the following. -Less than 4

Solve the problem. -A saleswoman packed 3 jackets and 6 skirts. With one jacket, she could wear all 6 skirts. With another jacket, she could wear 5 skirts. With the third jacket, she could wear only 2 skirts. How Many different combinations did she have?

Find the number of ways to get the following card combinations from a 52-card deck. -All tens in a five-card hand

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

Solve the problem. -How many ways can a president, vice-president, and secretary be chosen from a club with 12 members?

How many 10-digit telephone numbers (area code + number)are possible if the first digit cannot be zero, the first three digits cannot be 800 or 900, and the number must end in 0000?

Solve the problem. -Maria, Daniel, Stephanie, Michael, Elena, Tyler, Sue, and Dimitri have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if Maria And Daniel are to sit next to each other?

Of the 2,598,960 different five-card hands possible from a deck of 52 playing cards, how many would contain all black cards.

A pool of possible jurors consists of 15 men and 10 women. How many different juries consisting of 5 men and 7 women are possible?

Solve the problem. -A restaurant offered salads with 3 type(s)of dressing and 6 different topping(s). How many different types of salad could be ordered?

If two fair dice, one red and one white, are rolled, in how many ways can the result be obtained? -A different number on each die.

Solve the problem. -A baseball manager has 10 players of the same ability. How many 9 player starting lineups can he create?

Solve the problem. -How many 5-card poker hands consisting of 2 aces and 3 kings are possible with an ordinary 52-card deck?

Solve the problem. -A chef makes 3 kinds of fruit salad. Raisins, coconut, almonds, or marshmallows can be added to each kind of salad. How many different fruit salads are possible?

The library is to be given 10 books as a gift. The books will be selected from a list of 23 titles. If each book selected must have a different title, how many possible selections are there?

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

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

Filters

Exam 10: Counting Methods

Read each combination value directly from Pascal's Triangle. - 9C8{ } _ { 9 } \mathrm { C } _ { 8 }9​C8​

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result. -The sum of the numbers showing is 5.

Using the 36 possibilities found in the product table for rolling two dice, list and count the outcomes for which the sum (for both dice)is the following. -Less than 4

Solve the problem. -A saleswoman packed 3 jackets and 6 skirts. With one jacket, she could wear all 6 skirts. With another jacket, she could wear 5 skirts. With the third jacket, she could wear only 2 skirts. How Many different combinations did she have?

Find the number of ways to get the following card combinations from a 52-card deck. -All tens in a five-card hand

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

Solve the problem. -How many ways can a president, vice-president, and secretary be chosen from a club with 12 members?

How many 10-digit telephone numbers (area code + number)are possible if the first digit cannot be zero, the first three digits cannot be 800 or 900, and the number must end in 0000?

Solve the problem. -Maria, Daniel, Stephanie, Michael, Elena, Tyler, Sue, and Dimitri have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if Maria And Daniel are to sit next to each other?

Of the 2,598,960 different five-card hands possible from a deck of 52 playing cards, how many would contain all black cards.

A pool of possible jurors consists of 15 men and 10 women. How many different juries consisting of 5 men and 7 women are possible?

Solve the problem. -A restaurant offered salads with 3 type(s)of dressing and 6 different topping(s). How many different types of salad could be ordered?

If two fair dice, one red and one white, are rolled, in how many ways can the result be obtained? -A different number on each die.

Solve the problem. -A baseball manager has 10 players of the same ability. How many 9 player starting lineups can he create?

Solve the problem. -How many 5-card poker hands consisting of 2 aces and 3 kings are possible with an ordinary 52-card deck?

Solve the problem. -A chef makes 3 kinds of fruit salad. Raisins, coconut, almonds, or marshmallows can be added to each kind of salad. How many different fruit salads are possible?

The library is to be given 10 books as a gift. The books will be selected from a list of 23 titles. If each book selected must have a different title, how many possible selections are there?

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

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

Filters

Read each combination value directly from Pascal's Triangle. - ${ } _ { 9 } \mathrm { C } _ { 8 }$