Question 1

A school offers 4 music clubs and 3 sports clubs. A student may join 3 club. Which counting principle should be used to determine the total number of possible choices?

Accepted Answer

Multiplication Principle

Answer

Permutation Principle

Answer

Addition Principle

Answer

Combination Principle

Question 2

A cafeteria allows students to choose one main dish and one drink. Which counting principle is appropriate for finding the total number of meal choices?

Accepted Answer

Multiplication Principle

Answer

Addition Principle

Answer

Combination Principle

Answer

Permutation Principle

Question 3

A science teacher asks students to either:
arrange the top three winners in a quiz contest, or 
select three students to represent the class in a seminar. 
Which statement correctly identifies the counting methods?

Accepted Answer

The first uses a permutation while the second uses a combination.

Answer

Both situations use permutations.

Answer

The first uses a combination while the second uses a permutation.

Answer

Both situations use combinations.

Question 4

A school event includes the following activities:
I. Electing a president, vice president, and secretary
II. Choosing five volunteers for a clean-up drive
III. Awarding gold, silver, and bronze medals
Which activities require permutations?

Accepted Answer

I and III

Answer

I only

Answer

II only

Answer

II and III

Question 5

A class has 8 students competing for President, Vice President, and Secretary. How many different ways can these officers be chosen?

Accepted Answer

336

Answer

168

Answer

512

Answer

56

Question 6

A basketball coach must choose 5 players from 10 team members to start the game. How many different starting lineups are possible?

Accepted Answer

252

Answer

210

Answer

120

Answer

720

Question 7

A student claims that 10C3​ and 10C7​ always have the same value. Which explanation best justifies the claim?

Accepted Answer

Choosing 3 objects automatically determines the remaining 7 objects.

Answer

They use the same factorial formula.

Answer

Both expressions always equal 10.

Answer

They both involve permutations.

Question 8

A teacher asks students to determine whether it is easier to compute 15C13​ directly or use a property of combinations. Which reasoning is most appropriate?

Accepted Answer

Replace 15C13​ with 15C2​ because they are equal.

Answer

Compute 15P13​ first.

Answer

Use multiplication since order matters.

Answer

Divide 15 by 13 before applying the formula.

Question 9

A student solved the following problem:
"How many ways can a president, vice president, and secretary be selected from 8 students?"
The student used the solution: 8C3 = 56 
How would you evaluate the student's solution?

Accepted Answer

The solution is incorrect because permutations should be used since positions are different.

Answer

The solution is correct because combinations and permutations always produce the same answer.

Answer

The solution is correct because only three students are chosen.

Answer

The solution is incorrect because multiplication should be used instead of permutations.

Question 10

A teacher asks students to solve the following:
"Award gold, silver, and bronze medals to 12 finalists."
Which method is the best choice?

Accepted Answer

Permutation

Answer

Addition Principle

Answer

Combination

Answer

Multiplication Principle only

Question 11

How many distinct arrangements can be formed using the letters of the word BALLOON?

Accepted Answer

1260

Answer

420

Answer

5040

Answer

630

Question 12

Six friends are seated around a circular table. How many different seating arrangements are possible?

Accepted Answer

120

Answer

720

Answer

1440

Answer

360

Question 13

Eight different books are to be distributed equally among two students, with each student receiving four books. How many different distributions are possible?

Accepted Answer

70

Answer

35

Answer

56

Answer

140

Question 14

Five identical gift bags are to be distributed among three children. Which statement correctly describes this problem?

Accepted Answer

The objects are identical, so only the number of bags received matters.

Answer

The order in which the bags are distributed matters.

Answer

A permutation formula should always be used.

Answer

The objects are distinct.

Question 15

A student must choose one Mathematics elective from 4 options and one Science elective from 3 options. How many different schedules are possible?

Accepted Answer

12

Answer

36

Answer

7

Answer

24

Question 16

A school plans to form a committee consisting of 1 chairperson, 1 vice-chairperson, and 4 members from 12 teachers. Which sequence of counting techniques is most appropriate?

Accepted Answer

Use permutations for the officers, then combinations for the remaining members.

Answer

Use the Addition Principle because officers and members are different groups.

Answer

Use permutations for all six positions.

Answer

Use combinations for all positions.

Question 17

A card is drawn from cards numbered 1–10.
Event B = "drawing a number greater than 6."
Which set correctly represents Event B?

Accepted Answer

{7, 8, 9, 10}

Answer

{6, 7, 8, 9, 10}

Answer

{5, 6, 7, 8}

Answer

{8, 9, 10}

Question 18

A spinner has sections labeled:
Red, Blue, Green, Yellow
Event C = "landing on a warm color."
Which correctly represents Event C?

Accepted Answer

{Red, Yellow}

Answer

{Green}

Answer

{Blue, Green}

Answer

{Blue, Yellow}

Question 19

A student claims that the probability of drawing a king from a standard deck of cards is 1/13. How would you evaluate the student's answer?

Accepted Answer

Correct because there are four kings among fifty-two cards.

Answer

Correct because every suit has one king.

Answer

Incorrect because the probability is 1/4

Answer

Incorrect because the probability is 13/52

Question 20

A teacher solved a probability problem using permutations when selecting three committee members from ten students. How should this solution be evaluated?

Accepted Answer

Incorrect because combinations should be used.

Answer

Correct because both methods give the same probability.

Answer

Incorrect because the Addition Principle should be used.

Answer

Correct because selecting members requires order.

Question 21

A card is drawn.
Event A = drawing a queen
Event B = drawing a heart
Are these events mutually exclusive?

Accepted Answer

No, because the queen of hearts belongs to both events.

Answer

Yes, because there are four queens.

Answer

No, because every heart is a queen.

Answer

Yes, because every queen is a heart.

Question 22

A spinner has numbers 1–8.
Event A = landing on a multiple of 2
Event B = landing on a multiple of 4
These events are

Accepted Answer

not mutually exclusive.

Answer

impossible.

Answer

complements.

Answer

mutually exclusive.

Question 23

In a class,
18 students play basketball,
15 play volleyball,
5 play both.
A student computes 18 + 15 = 33 
Which evaluation is correct?

Accepted Answer

Incorrect because the overlap must be subtracted.

Answer

Correct because sports are independent.

Answer

Incorrect because multiplication should be used.

Answer

Correct because all students are counted.

Question 24

Which student's reasoning correctly explains why the overlap is subtracted when finding P(A∪B)?

Accepted Answer

It removes outcomes counted twice.

Answer

It guarantees independence.

Answer

It avoids multiplying probabilities.

Answer

It makes the answer smaller than both probabilities.

Question 25

Which situation involves conditional probability?

Accepted Answer

Drawing a second card without replacement after the first card is an ace.

Answer

Drawing one card from a full deck.

Answer

Spinning a spinner once.

Answer

Rolling a die and getting 5.

Question 26

Which characteristic distinguishes conditional probability from simple probability?

Accepted Answer

It changes the sample space because another event has occurred.

Answer

It is always larger than simple probability.

Answer

It assumes all outcomes are equally likely.

Answer

It considers only one event.

Question 27

A box contains 5 red and 3 blue balls. One red ball is drawn and not replaced. A student claims the probability that the second ball is blue is 3/8. How should the solution be evaluated?

Accepted Answer

Incorrect because it should be 3/7

Answer

Incorrect because it should be 2/7

Answer

Correct

Answer

Correct because replacement is assumed.

Question 28

A student solved a conditional probability problem by using the original sample space even after one card had already been removed. What error did the student make?

Accepted Answer

Failed to adjust the sample space after the first event.

Answer

Ignored mutually exclusive events.

Answer

Used the complement rule incorrectly.

Answer

Used combinations instead of permutations.

Question 29

Which pair of events is independent?

Accepted Answer

Tossing a coin and rolling a die.

Answer

Drawing two cards without replacement.

Answer

Choosing two marbles from a bag without replacement.

Answer

Selecting two students from a class without replacement.

Question 30

A teacher says, "If the occurrence of Event A changes the probability of Event B, then the events are independent." How should this statement be analyzed?

Accepted Answer

Incorrect because independent events do not affect each other.

Answer

Correct because probabilities always change.

Answer

Correct for combinations only.

Answer

Correct only for mutually exclusive events.

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