Question 1

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

Accepted Answer

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

Question 2

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

Question 3

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 7 and one die is a 3.

Accepted Answer

A) (3,4) 
B) (4,3) 
C) 0 
D) (3,4),(4,3) 
A) (3,4) 
B) (4,3) 
C) 0 
D) (3,4),(4,3)

Question 4

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.
-Rotate 90° clockwise
 $\begin{array}{|c|c|c|}
\hline 12 & 14 & 4 \
\hline 2 & 10 & 18 \
\hline 16 & 6 & 8 \
\hline
\end{array}$

Accepted Answer

A) 
$\begin{array}{|c|c|c|}
\hline 4 & 18 & 8 \
\hline 14 & 10 & 6 \
\hline 12 & 2 & 16 \
\hline
\end{array}$ 
B) 
$\begin{array}{|c|c|c|}
\hline 8 & 6 & 16 \
\hline 18 & 10 & 2 \
\hline 4 & 14 & 12 \
\hline
\end{array}$ 
C) 
$\begin{array}{|c|c|c|}
\hline 12 & 14 & 4 \
\hline 2 & 10 & 18 \
\hline 16 & 6 & 8 \
\hline
\end{array}$ 
D) 
$\begin{array}{|c|c|c|}
\hline 16 & 2 & 12 \
\hline 6 & 10 & 14 \
\hline 8 & 18 & 4 \
\hline
\end{array}$ 
A) 
$\begin{array}{|c|c|c|}
\hline 4 & 18 & 8 \
\hline 14 & 10 & 6 \
\hline 12 & 2 & 16 \
\hline
\end{array}$ 
B) 
$\begin{array}{|c|c|c|}
\hline 8 & 6 & 16 \
\hline 18 & 10 & 2 \
\hline 4 & 14 & 12 \
\hline
\end{array}$ 
C) 
$\begin{array}{|c|c|c|}
\hline 12 & 14 & 4 \
\hline 2 & 10 & 18 \
\hline 16 & 6 & 8 \
\hline
\end{array}$ 
D) 
$\begin{array}{|c|c|c|}
\hline 16 & 2 & 12 \
\hline 6 & 10 & 14 \
\hline 8 & 18 & 4 \
\hline
\end{array}$

Question 5

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

Accepted Answer

A) hhhh, hhht, hhth, hthh, hhtt, tthh, ttht, ttth, tttt 
B) hhhh, hhht, hhth, hhtt, tthh, ttht, ttth, tttt 
C) hhhh, hhht, hhtt, tthh, ttht, ttth, tttt 
D) hhhh, hhht, hhth, hhtt 
A) hhhh, hhht, hhth, hthh, hhtt, tthh, ttht, ttth, tttt 
B) hhhh, hhht, hhth, hhtt, tthh, ttht, ttth, tttt 
C) hhhh, hhht, hhtt, tthh, ttht, ttth, tttt 
D) hhhh, hhht, hhth, hhtt

Question 6

Find the number of ways to get the following card combinations from a 52-card deck.
-No face cards in a five-card hand

Accepted Answer

A) 658,008 ways 
B) 319,865 ways 
C) 127,946 ways 
D) 639,730 ways 
A) 658,008 ways 
B) 319,865 ways 
C) 127,946 ways 
D) 639,730 ways

Question 7

Evaluate the expression.
-Determine the number of combinations of 7 things taken 4 at a time.

Accepted Answer

A) 210 
B) 12 
C) 35 
D) 420 
A) 210 
B) 12 
C) 35 
D) 420

Question 8

Solve the problem.
-Four married couples have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if all the women sit together and all the men sit
Together?

Accepted Answer

A) 48 
B) 256 
C) 576 
D) 1152 
A) 48 
B) 256 
C) 576 
D) 1152

Question 9

Evaluate the factorial expression.
-$\frac { 7 ! } { 6 ! }$

Accepted Answer

A)  7 
B)  $\frac { 7 } { 6 }$ 
C)  $7 !$ 
D)  1 
A)  7 
B)  $\frac { 7 } { 6 }$ 
C)  $7 !$ 
D)  1

Question 10

Given a group of students: G = {Allen, Brenda, Chad, Dorothy, Eric} or G = {A, B, C, D, E}, list and count the different ways of choosing the following officers or representatives for student congress. Assume that no one can hold more than one office.
-A president, a secretary, and a treasurer, if the president must be a woman and the other two must be men

Accepted Answer

A) BAC, BAE, DAC, DAE; 4 
B) BAC, BAE, BCE, DAC, DAE, DCE, BCA, BEA, BEC, DCA, DEA, DEC; 12 
C) BAC, BAE, BCE, DAC, DAE, DCE; 6 
D) ABD, CBD, EBD; 3 
A) BAC, BAE, DAC, DAE; 4 
B) BAC, BAE, BCE, DAC, DAE, DCE, BCA, BEA, BEC, DCA, DEA, DEC; 12 
C) BAC, BAE, BCE, DAC, DAE, DCE; 6 
D) ABD, CBD, EBD; 3

Question 11

License plates are made using 3 letters followed by 3 digits. How many plates can be made if repetition of letters and digits is allowed?

Accepted Answer

A) 1,000,000 
B) 308,915,776 
C) 1,757,600 
D) 17,576,000 
A) 1,000,000 
B) 308,915,776 
C) 1,757,600 
D) 17,576,000

Question 12

Evaluate the factorial expression.
-$\frac { 7 ! } { 5 ! 2 ! }$

Accepted Answer

A)  21 
B)  42 
C)  7 
D)  1 
A)  21 
B)  42 
C)  7 
D)  1

Question 13

Four married couples have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if all the women sit together and all the men sit
Together?

Accepted Answer

A) 48 
B) 256 
C) 576 
D) 1152 
A) 48 
B) 256 
C) 576 
D) 1152

Question 14

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

Accepted Answer

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

Question 15

If two fair dice, one red and one white, are rolled, in how many ways can the result be obtained?
-The sum of the numbers showing is either 3 or 4.

Accepted Answer

A) 4 ways 
B) 5 ways 
C) 2 ways 
D) 3 ways 
A) 4 ways 
B) 5 ways 
C) 2 ways 
D) 3 ways

Question 16

If two fair dice, one red and one white, are rolled, in how many ways can the result be obtained?
-The sum of the two dice is less than 5.

Accepted Answer

A) 10 ways 
B) 5 ways 
C) 4 ways 
D) 6 ways 
A) 10 ways 
B) 5 ways 
C) 4 ways 
D) 6 ways

Question 17

Solve the problem.
-If a given set has nine elements, how many of its subsets have at least five elements?

Accepted Answer

A) 256 subsets 
B) 130 subsets 
C) 32 subsets 
D) 255 subsets 
A) 256 subsets 
B) 130 subsets 
C) 32 subsets 
D) 255 subsets

Question 18

Three noncollinear points determine a triangle. How many different triangles are determined by 7 points in a plane, no three of which are collinear?

Accepted Answer

A) 21 
B) 840 
C) 210 
D) 35 
A) 21 
B) 840 
C) 210 
D) 35

Question 19

Given a group of students: G = {Allen, Brenda, Chad, Dorothy, Eric} or G = {A, B, C, D, E}, list and count the different ways of choosing the following officers or representatives for student congress. Assume that no one can hold more than one office.
-A male president and three representatives

Accepted Answer

A) ABCD, ACDE, CABD, CABE, EABC, EABD; 6 
B) ABCD, CABD, CABE, EABD, ADCB, AEDC, CDBA, ECBA, EDBA; 9 
C) ABCD, CABD, EABC; 3 
D) ABCD, ACDE, ABCE, ABDE, CABD, CABE, CBDE, CADE, EABC, EABD, EBCD, EACD; 12 
A) ABCD, ACDE, CABD, CABE, EABC, EABD; 6 
B) ABCD, CABD, CABE, EABD, ADCB, AEDC, CDBA, ECBA, EDBA; 9 
C) ABCD, CABD, EABC; 3 
D) ABCD, ACDE, ABCE, ABDE, CABD, CABE, CBDE, CADE, EABC, EABD, EBCD, EACD; 12

Question 20

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

Accepted Answer

A) 15 ways 
B) 32 ways 
C) 16 ways 
D) 31 ways 
A) 15 ways 
B) 32 ways 
C) 16 ways 
D) 31 ways

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

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?

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 7 and one die is a 3.

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

Find the number of ways to get the following card combinations from a 52-card deck. -No face cards in a five-card hand

Evaluate the expression. -Determine the number of combinations of 7 things taken 4 at a time.

Solve the problem. -Four married couples have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if all the women sit together and all the men sit Together?

Evaluate the factorial expression. - $\frac { 7 ! } { 6 ! }$

License plates are made using 3 letters followed by 3 digits. How many plates can be made if repetition of letters and digits is allowed?

Evaluate the factorial expression. - $\frac { 7 ! } { 5 ! 2 ! }$

Four married couples have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if all the women sit together and all the men sit Together?

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

If two fair dice, one red and one white, are rolled, in how many ways can the result be obtained? -The sum of the numbers showing is either 3 or 4.

If two fair dice, one red and one white, are rolled, in how many ways can the result be obtained? -The sum of the two dice is less than 5.

Solve the problem. -If a given set has nine elements, how many of its subsets have at least five elements?

Three noncollinear points determine a triangle. How many different triangles are determined by 7 points in a plane, no three of which are collinear?

Given a group of students: G = {Allen, Brenda, Chad, Dorothy, Eric} or G = {A, B, C, D, E}, list and count the different ways of choosing the following officers or representatives for student congress. Assume that no one can hold more than one office. -A male president and three representatives

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

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

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

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?

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 7 and one die is a 3.

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

Find the number of ways to get the following card combinations from a 52-card deck. -No face cards in a five-card hand

Evaluate the expression. -Determine the number of combinations of 7 things taken 4 at a time.

Solve the problem. -Four married couples have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if all the women sit together and all the men sit Together?

Evaluate the factorial expression. - 7!6!\frac { 7 ! } { 6 ! }6!7!​

License plates are made using 3 letters followed by 3 digits. How many plates can be made if repetition of letters and digits is allowed?

Evaluate the factorial expression. - 7!5!2!\frac { 7 ! } { 5 ! 2 ! }5!2!7!​

Four married couples have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if all the women sit together and all the men sit Together?

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

If two fair dice, one red and one white, are rolled, in how many ways can the result be obtained? -The sum of the numbers showing is either 3 or 4.

If two fair dice, one red and one white, are rolled, in how many ways can the result be obtained? -The sum of the two dice is less than 5.

Solve the problem. -If a given set has nine elements, how many of its subsets have at least five elements?

Three noncollinear points determine a triangle. How many different triangles are determined by 7 points in a plane, no three of which are collinear?

Given a group of students: G = {Allen, Brenda, Chad, Dorothy, Eric} or G = {A, B, C, D, E}, list and count the different ways of choosing the following officers or representatives for student congress. Assume that no one can hold more than one office. -A male president and three representatives

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

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

Evaluate the factorial expression. - $\frac { 7 ! } { 6 ! }$

Evaluate the factorial expression. - $\frac { 7 ! } { 5 ! 2 ! }$