Combinatorics: Master the Art of Counting and Arranging

Question 1

A restaurant offers 6 different appetizers, 8 different main courses, and 4 different desserts. How many different 3-course meals are possible?

Accepted Answer

192

Answer

144

Answer

96

Answer

48

Question 2

A lottery has 50 balls numbered from 1 to 50. If 6 balls are drawn without replacement, what is the probability of selecting the first 6 numbers in order?

Accepted Answer

1/15890700

Answer

1/50

Answer

1/120

Answer

1/220

Question 3

In a club with 15 members, a committee of 5 people is to be formed. How many different committees are possible?

Accepted Answer

3003

Answer

3030

Answer

455

Answer

1050

Question 4

A group of 8 students is to be arranged in a row for a photograph. How many different arrangements are possible?

Accepted Answer

40320

Answer

4096

Answer

8192

Answer

16128

Question 5

A bag contains 5 red marbles, 3 green marbles, and 2 blue marbles. If 2 marbles are drawn at random with replacement, what is the probability of drawing a red marble and then a blue marble?

Accepted Answer

0.24

Answer

0.30

Answer

0.40

Answer

0.15

Question 6

A combination lock has 4 dials, each of which can be set to any of the 10 digits from 0 to 9. What is the total number of possible combinations?

Accepted Answer

10000

Answer

400

Answer

4000

Answer

1000

Question 7

How many different 6-letter passwords can be created if each character can be any lowercase letter from 'a' to 'z'?

Accepted Answer

26^6

Answer

6^26

Answer

26^5

Answer

5^26

Question 8

A survey asks participants to select their favorite color from a list of 5 colors. How many different ways can a participant rank their top 3 favorite colors?

Accepted Answer

60

Answer

20

Answer

120

Answer

30

Question 9

There are 6 different books to be arranged on a shelf. How many distinct arrangements are possible?

Accepted Answer

720

Answer

360

Answer

120

Answer

144

Question 10

A survey asks respondents to choose their favorite subject from a list of 5 subjects and their favorite sport from a list of 4 sports. How many different ways can the respondents answer?

Accepted Answer

20

Answer

30

Answer

10

Answer

25

Question 11

A restaurant offers 5 appetizers, 8 main courses, and 3 desserts. How many different three-course meals can be ordered?

Accepted Answer

120

Answer

16

Answer

60

Question 12

What is the value of ₇P₃?

Accepted Answer

210

Answer

35

Answer

5040

Question 13

In how many ways can 5 different books be arranged on a shelf?

Accepted Answer

120

Answer

24

Answer

25

Answer

5

Question 14

How many 4-letter codes can be formed using the letters A, B, C, D, and E if repetition of letters is allowed?

Accepted Answer

625

Answer

20

Answer

24

Answer

120

Question 15

A bag contains 6 red marbles and 4 blue marbles. How many ways can 2 red marbles and 1 blue marble be drawn?

Accepted Answer

60

Answer

10

Answer

24

Answer

15

Question 16

What is the coefficient of the x³y² term in the expansion of (x + y)⁵?

Accepted Answer

10

Answer

5

Answer

1

Answer

20

Question 17

How many different 5-card hands can be dealt from a standard deck of 52 cards, if the order of the cards in the hand does not matter?

Accepted Answer

2,598,960

Answer

120

Answer

311,875,200

Answer

260

Question 18

A group of 8 friends are going to a concert.  If only 5 tickets are available, how many different groups of friends can go?

Accepted Answer

56

Answer

120

Answer

336

Answer

40,320

Question 19

In a club with 25 members, how many different ways can a president, vice president, and treasurer be chosen?

Accepted Answer

13260

Answer

2500

Answer

1250

Answer

3250

Question 20

A committee of 5 members is to be formed from a group of 10 people. How many different committees are possible if John and Mary must be on the committee?

Accepted Answer

120

Answer

252

Answer

360

Question 21

A lock has 4 dials, each with 10 digits (0-9). How many different 4-digit combinations are possible?

Accepted Answer

10000

Answer

2500

Answer

1000

Answer

4000

Question 22

In a class of 25 students, how many ways can a teacher select 5 students to form a group?

Accepted Answer

53130

Answer

62500

Answer

1225

Answer

125

Question 23

A password consists of 3 lowercase letters followed by 2 digits. How many possible passwords can be created?

Accepted Answer

15600

Answer

7800

Answer

26000

Question 24

In a group of 5 friends, how many different pairs of friends can be formed to go to the cinema?

Accepted Answer

10

Answer

15

Answer

20

Answer

5

Question 25

A restaurant offers 3 appetizers, 5 main courses, and 2 desserts. How many different 3-course meals can a customer choose from?

Accepted Answer

30

Answer

60

Answer

15

Answer

10

Question 26

In how many ways can 5 students be arranged in a line for a photograph?

Accepted Answer

120

Answer

25

Answer

720

Answer

60

Question 27

A committee of 3 members is to be chosen from a group of 8 people. How many different committees can be formed?

Accepted Answer

56

Answer

3360

Answer

336

Answer

24

Question 28

How many different ways can the letters in the word 'MATH' be arranged?

Accepted Answer

24

Answer

4

Answer

12

Answer

6

Question 29

A bag contains 5 red balls, 3 blue balls, and 2 green balls. If 2 balls are drawn randomly without replacement, what is the probability that both are red?

Accepted Answer

2/7

Answer

1/7

Answer

1/2

Answer

5/10

Question 30

A store offers 6 different flavors of ice cream and 4 different toppings. How many different single-scoop ice cream cones with one topping can be made?

Accepted Answer

24

Answer

48

Answer

15

Answer

10

Question 31

A group of 10 friends are going to a movie. In how many ways can they be seated in a row if 2 specific friends want to sit next to each other?

Accepted Answer

1440

Answer

180

Answer

360

Answer

720

Question 32

A team of 4 people needs to be chosen from a group of 10 people. How many different teams can be formed if there is no specific role or order within the team?

Accepted Answer

210

Answer

40

Answer

100

Answer

5040

Question 33

A code consists of 3 letters followed by 2 digits. How many different codes are possible if repetition of letters and digits is allowed?

Accepted Answer

1757600

Answer

17576

Answer

175760

Answer

17576000

Question 34

A store sells 5 types of cookies. A customer wants to buy 3 cookies. How many different combinations of cookies can the customer choose?

Accepted Answer

10

Answer

15

Answer

25

Answer

20