Question 1

Evaluate ${ } _ { n } C _ { r }$ using the formula from this section.? ${ } _ { 6 } C _ { 1 }$ ?

Accepted Answer

A) 8 
B) 7 
C) 9 
D) 10 
E) 6 
A) 8 
B) 7 
C) 9 
D) 10 
E) 6 E

Question 2

In how many ways can seven children posing for a photograph line up in a row? ​

Accepted Answer

A) 5,040 
B) 5,140 
C) 5,090 
D) 6,040 
E) 5,540 
A) 5,040 
B) 5,140 
C) 5,090 
D) 6,040 
E) 5,540 A

Question 3

Twelve weightlifters are competing in the dead-lift competition.In how many ways can the weightlifters finish first,second,and third (no ties)?

Accepted Answer

A) 1,728 
B) 6 
C) 36 
D) 3 
E) 1,320 
A) 1,728 
B) 6 
C) 36 
D) 3 
E) 1,320 E

Question 4

Evaluate the expression.? $\left( \begin{array} { l } 
96 \
95
\end{array} \right)$ ?

Accepted Answer

A) 86 
B) 1 
C) 96 
D) 95 
E) -9 
A) 86 
B) 1 
C) 96 
D) 95 
E) -9

Question 5

Eight sprinters have qualified for the finals in the 100-meter dash at the NCAA national track meet.In how many ways can the sprinters come in first,second,and third? (Assume there are no ties. ) ​

Accepted Answer

A) 486 
B) 386 
C) 436 
D) 336 
E) 536 
A) 486 
B) 386 
C) 436 
D) 336 
E) 536

Question 6

Evaluate ${ } _ { n } C _ { r }$ using the formula from this section.? ${ } _ { 5 } C _ { 3 }$ ?

Accepted Answer

A) 84 
B) 60 
C) 10 
D) 20 
E) 35 
A) 84 
B) 60 
C) 10 
D) 20 
E) 35

Question 7

A college needs two additional faculty members: a chemist and a statistician.In how many ways can these positions be filled if there are three applicants for the chemistry position and four applicants for the statistics position?

Accepted Answer

A) 10 
B) 11 
C) 16 
D) 14 
E) 12 
A) 10 
B) 11 
C) 16 
D) 14 
E) 12

Question 8

Evaluate ${ } _ { n } P _ { r }$ .? ${ } _ { 7 } P _ { 4 }$ ?

Accepted Answer

A) 5,040 
B) 1,680 
C) 3,024 
D) 35 
E) 840 
A) 5,040 
B) 1,680 
C) 3,024 
D) 35 
E) 840

Question 9

Evaluate ${ } _ { n } P _ { r }$ .? $${ } _ { 2 } P _ { 2 }$$ ?

Accepted Answer

A) 2 
B) 20 
C) 30 
D) 6 
E) 12 
A) 2 
B) 20 
C) 30 
D) 6 
E) 12

Question 10

A customer can choose one of two amplifiers,one of three compact disc players,and one of five speaker models for an entertainment system.Determine the number of possible system configurations. ​

Accepted Answer

A) 40 
B) 35 
C) ​30 
D) 25 
E) 20 
A) 40 
B) 35 
C) ​30 
D) 25 
E) 20

Question 11

How many three-digit numbers can be formed under the following condition? ​
The leading digit cannot be zero and the number must be even.
​

Accepted Answer

A) 900 
B) 450 
C) 350 
D) 400 
E) 550 
A) 900 
B) 450 
C) 350 
D) 400 
E) 550

Question 12

How many three-digit numbers can be formed under the following condition? ​
The leading digit cannot be zero.
​

Accepted Answer

A) 800 
B) ​500 
C) ​900 
D) ​600 
E) ​700 
A) 800 
B) ​500 
C) ​900 
D) ​600 
E) ​700

Question 13

Evaluate ${ } _ { n } P _ { r }$ .? ${}_{100} P _ { 3 }$ ?

Accepted Answer

A) 94,109,400 
B) 161,700 
C) 970,200 
D) 858,277,728,000 
E) 9,034,502,400 
A) 94,109,400 
B) 161,700 
C) 970,200 
D) 858,277,728,000 
E) 9,034,502,400

Question 14

There are six processes involved in assembling a product,and these processes can be performed in any order.The management wants to test each order to determine which is the least time-consuming.How many different orders will have to be tested? ​

Accepted Answer

A) 1,220 
B) 720 
C) 770 
D) 820 
E) 1,720 
A) 1,220 
B) 720 
C) 770 
D) 820 
E) 1,720

Question 15

Evaluate ${ } _ { n } C _ { r }$ using the formula from this section.? ${ } _ { 25 } C _ { 4 }$ ?

Accepted Answer

A) 177,100 
B) 303,600 
C) 480,700 
D) 12,650 
E) 53,130 
A) 177,100 
B) 303,600 
C) 480,700 
D) 12,650 
E) 53,130

Question 16

Ten people are boarding an aircraft.Two have tickets for first class and board before those in the economy class.In how many ways can the ten people board the aircraft? ​

Accepted Answer

A) 240 
B) 48 
C) 80,640 
D) 12 
E) 1,440 
A) 240 
B) 48 
C) 80,640 
D) 12 
E) 1,440

Question 17

Determine the number of ways a computer can randomly generate an integer between 10 and 20 (inclusive)that is a composite number (a composite number is a number that is not a prime number).

Accepted Answer

A) ​8 
B) ​6 
C) 7 
D) 6 
E) 9 
A) ​8 
B) ​6 
C) 7 
D) 6 
E) 9

Question 18

Find the number of diagonals of a dodecagon (12 sides).(A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon. )

Accepted Answer

A) 24 
B) 12 
C) 144 
D) 132 
E) 54 
A) 24 
B) 12 
C) 144 
D) 132 
E) 54

Question 19

Determine the number of ways a computer can randomly generate the even integers from 1 through 30. ​

Accepted Answer

A) 17 
B) 13 
C) 15 
D) 11 
E) 19 
A) 17 
B) 13 
C) 15 
D) 11 
E) 19

Question 20

A college student is preparing a course schedule for the next semester.The student may select one of five mathematics courses,one of two science courses,and one of five courses from the social sciences and humanities.How many schedules are possible? ​

Accepted Answer

A) 55 
B) 45 
C) 60 
D) 50 
E) 40 
A) 55 
B) 45 
C) 60 
D) 50 
E) 40

Exam 57: Counting Principles

Evaluate nCr{ } _ { n } C _ { r }n​Cr​ using the formula from this section.? 6C1{ } _ { 6 } C _ { 1 }6​C1​ ?

In how many ways can seven children posing for a photograph line up in a row? ​

Twelve weightlifters are competing in the dead-lift competition.In how many ways can the weightlifters finish first,second,and third (no ties)?

Evaluate the expression.? (9695)\left( \begin{array} { l } 96 \\95\end{array} \right)(9695​) ?

Eight sprinters have qualified for the finals in the 100-meter dash at the NCAA national track meet.In how many ways can the sprinters come in first,second,and third? (Assume there are no ties. ) ​

Evaluate nCr{ } _ { n } C _ { r }n​Cr​ using the formula from this section.? 5C3{ } _ { 5 } C _ { 3 }5​C3​ ?

A college needs two additional faculty members: a chemist and a statistician.In how many ways can these positions be filled if there are three applicants for the chemistry position and four applicants for the statistics position?

Evaluate nPr{ } _ { n } P _ { r }n​Pr​ .? 7P4{ } _ { 7 } P _ { 4 }7​P4​ ?

Evaluate nPr{ } _ { n } P _ { r }n​Pr​ .? 2P2{ } _ { 2 } P _ { 2 }2​P2​ ?

A customer can choose one of two amplifiers,one of three compact disc players,and one of five speaker models for an entertainment system.Determine the number of possible system configurations. ​

How many three-digit numbers can be formed under the following condition? ​ The leading digit cannot be zero and the number must be even. ​

How many three-digit numbers can be formed under the following condition? ​ The leading digit cannot be zero. ​

Evaluate nPr{ } _ { n } P _ { r }n​Pr​ .? 100P3{}_{100} P _ { 3 }100​P3​ ?

There are six processes involved in assembling a product,and these processes can be performed in any order.The management wants to test each order to determine which is the least time-consuming.How many different orders will have to be tested? ​

Evaluate nCr{ } _ { n } C _ { r }n​Cr​ using the formula from this section.? 25C4{ } _ { 25 } C _ { 4 }25​C4​ ?

Ten people are boarding an aircraft.Two have tickets for first class and board before those in the economy class.In how many ways can the ten people board the aircraft? ​

Determine the number of ways a computer can randomly generate an integer between 10 and 20 (inclusive)that is a composite number (a composite number is a number that is not a prime number).

Find the number of diagonals of a dodecagon (12 sides).(A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon. )

Determine the number of ways a computer can randomly generate the even integers from 1 through 30. ​

A college student is preparing a course schedule for the next semester.The student may select one of five mathematics courses,one of two science courses,and one of five courses from the social sciences and humanities.How many schedules are possible? ​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Probability

Lines

Introduction to Conics Parabolas

Evaluate ${ } _ { n } C _ { r }$ using the formula from this section.? ${ } _ { 6 } C _ { 1 }$ ?

In how many ways can seven children posing for a photograph line up in a row?

Evaluate the expression.? $\left( \begin{array} { l } 96 \\95\end{array} \right)$ ?

Eight sprinters have qualified for the finals in the 100-meter dash at the NCAA national track meet.In how many ways can the sprinters come in first,second,and third? (Assume there are no ties. )

Evaluate ${ } _ { n } C _ { r }$ using the formula from this section.? ${ } _ { 5 } C _ { 3 }$ ?

Evaluate ${ } _ { n } P _ { r }$ .? ${ } _ { 7 } P _ { 4 }$ ?

Evaluate ${ } _ { n } P _ { r }$ .? ${ } _ { 2 } P _ { 2 }$ ?

A customer can choose one of two amplifiers,one of three compact disc players,and one of five speaker models for an entertainment system.Determine the number of possible system configurations.

How many three-digit numbers can be formed under the following condition? The leading digit cannot be zero and the number must be even.

How many three-digit numbers can be formed under the following condition? The leading digit cannot be zero.

Evaluate ${ } _ { n } P _ { r }$ .? ${}_{100} P _ { 3 }$ ?

There are six processes involved in assembling a product,and these processes can be performed in any order.The management wants to test each order to determine which is the least time-consuming.How many different orders will have to be tested?

Evaluate ${ } _ { n } C _ { r }$ using the formula from this section.? ${ } _ { 25 } C _ { 4 }$ ?

Ten people are boarding an aircraft.Two have tickets for first class and board before those in the economy class.In how many ways can the ten people board the aircraft?

Determine the number of ways a computer can randomly generate the even integers from 1 through 30.

A college student is preparing a course schedule for the next semester.The student may select one of five mathematics courses,one of two science courses,and one of five courses from the social sciences and humanities.How many schedules are possible?