Master Combinatorics: Counting Techniques for Mathematics Education

Question 1

A class has 12 boys and 15 girls. A committee of 5 is to be formed with 3 boys and 2 girls. How many different committees are possible?

Accepted Answer

1,320

Answer

2,300

Answer

1,760

Answer

2,730

Question 2

What is the coefficient of (x^5) in the expansion of ((x+2)^7)?

Accepted Answer

84

Answer

70

Answer

98

Answer

112

Question 3

An ice cream store offers 12 different flavors. How many distinct combinations of 4 different flavors can a customer choose for a cone?

Accepted Answer

1980

Answer

495

Answer

990

Answer

3960

Question 4

6 people are standing in a line for a photograph. How many different arrangements of people in the line are possible?

Accepted Answer

720

Answer

6

Answer

24

Answer

5040

Question 5

Expand the binomial expression: (2x + 3y)^3

Accepted Answer

8x^3 + 24x^2y + 36xy^2 + 27y^3

Answer

8x^3 + 12x^2y + 18xy^2 + 27y^3

Answer

8x^3 + 36x^2y + 54xy^2 + 27y^3

Answer

8x^3 + 12x^2y + 24xy^2 + 27y^3

Question 6

Determine the number of distinct arrangements of the letters in the word 'MISSISSIPPI'.

Accepted Answer

(11!)/(4!4!2!)

Answer

11!

Answer

11^4

Answer

4!4!2!

Question 7

How many different committees of 5 people can be formed from a group of 12 people?

Accepted Answer

792

Answer

1008

Answer

560

Question 8

How many ways can 6 different textbooks be arranged on a bookshelf?

Accepted Answer

720

Answer

360

Answer

120

Answer

180

Question 9

How many possible sums can be formed by rolling two dice?

Accepted Answer

36

Answer

18

Answer

21

Answer

24

Question 10

In a row of 5 chairs, how many distinct ways can 5 students be seated?

Accepted Answer

120

Answer

60

Answer

180

Question 11

A teacher has 6 different math problems and 4 different science problems. In how many ways can the teacher select 3 math problems and 2 science problems for a quiz?

Accepted Answer

240

Answer

480

Answer

360

Answer

120

Question 12

A bag contains 6 red balls and 4 blue balls. How many different ways are there to draw 3 balls from the bag without replacement?

Accepted Answer

60

Answer

240

Answer

30

Answer

120

Question 13

A group of 6 friends want to take a photo together. How many distinct arrangements are possible if they line up in a row?

Accepted Answer

720

Answer

360

Answer

120

Question 14

A group of 8 people is standing in a line. How many distinct arrangements are possible?

Accepted Answer

40,320

Answer

16,128

Answer

8,064

Answer

32,256

Question 15

A bag contains 4 red marbles, 3 blue marbles, and 5 green marbles. If 3 marbles are drawn at random with replacement, what is the probability of drawing 2 red marbles and 1 blue marble? Express your answer in its simplest form.

Accepted Answer

12/65

Answer

1/6

Answer

2/5

Answer

1/2

Question 16

In a race with 8 runners, in how many ways can the first three positions be filled?

Accepted Answer

56

Answer

24

Answer

36

Answer

48

Question 17

A committee of 5 people is to be selected from a group of 12. How many different committees are possible?

Accepted Answer

792

Answer

120

Answer

360

Answer

504

Question 18

In a survey, 50 people were asked their favorite color. The results were: 20 chose blue, 15 chose red, 10 chose green, and 5 chose yellow. What is the probability that a randomly selected person chose either blue or green?

Accepted Answer

0.60

Answer

0.40

Answer

0.30

Answer

0.70

Question 19

The binomial expansion of (x + y)^5 is:

Accepted Answer

x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5

Answer

x^5 - 5x^4y + 5x^3y^2 + 5x^2y^3 + 5xy^4 + y^5

Answer

x^5 - 5x^4y + 10x^3y^2 - 10x^2y^3 - 5xy^4 + y^5

Answer

x^5 + 5x^4y + 5x^3y^2 + 5x^2y^3 + 5xy^4 + y^5

Question 20

What is the coefficient of the x^3y^2 term in the expansion of (2x + 3y)^5?

Accepted Answer

180

Answer

108

Answer

90

Answer

216

Question 21

In a club with 10 members, how many different ways can a president, vice president, and secretary be chosen?

Accepted Answer

720

Answer

504

Answer

840

Answer

960

Question 22

A school cafeteria offers 4 main courses, 3 side dishes, and 2 desserts. How many different complete meals are possible?

Accepted Answer

24

Answer

12

Answer

36

Answer

48

Question 23

A box contains 6 red balls and 4 blue balls. If 3 balls are drawn at random without replacement, what is the probability of drawing 2 red balls and 1 blue ball?

Accepted Answer

0.36

Answer

0.24

Answer

0.60

Answer

0.48

Question 24

Which of the following is the binomial expansion of (x + 2)^3?

Accepted Answer

x³ + 6x² + 12x + 8

Answer

x³ + 8x² + 12x + 16

Answer

2x³ + 6x² + 12x + 8

Answer

x³ + 4x² + 6x + 4

Question 25

How many terms are in the expansion of (2x - 3y)^5?

Accepted Answer

6

Answer

7

Answer

5

Answer

8

Question 26

Evaluate the expression 7C4:

Accepted Answer

35

Answer

28

Answer

42

Question 27

A bag contains 5 red and 3 blue marbles. If 2 marbles are drawn at random without replacement, what is the probability of drawing a red marble and then a blue marble?

Accepted Answer

0.3

Answer

0.4

Answer

0.2

Answer

0.5

Question 28

How many different ways can 6 students be seated in a row of 6 chairs?

Accepted Answer

720

Answer

120

Answer

360

Answer

240

Question 29

Evaluate the binomial coefficient C(5, 2).

Accepted Answer

10

Answer

20

Answer

25

Answer

15

Question 30

Use the binomial theorem to expand (x - 2)^3.

Accepted Answer

x^3 - 6x^2 + 12x - 8

Answer

x^3 - 8x^2 + 6x - 12

Answer

x^3 - 8x^2 + 12x - 6

Answer

x^3 - 6x^2 + 8x - 12

Question 31

A bag contains 5 red balls and 7 blue balls. If 2 balls are drawn from the bag without replacement, what is the probability of drawing 1 red ball and 1 blue ball?

Accepted Answer

35/84

Answer

25/84

Answer

55/84

Answer

45/84

Question 32

A password consists of 4 digits, each of which can be from 0 to 9. How many different passwords are possible?

Accepted Answer

10000

Answer

12000

Answer

4000

Answer

8000

Question 33

A box contains 6 different books. In how many ways can 3 books be selected from the box and arranged on a shelf?

Accepted Answer

120

Answer

240

Answer

180

Answer

60