Free AMC 12 Combinatorics & Probability Questions and Answers — Questions and Answers

Question 1: How many ways can you arrange the letters of the word 'MATH'?

24 (Correct answer)
12
16
8

Correct answer: 24

The number of ways to arrange n distinct items is given by n! (n factorial). The word 'MATH' has 4 distinct letters. Therefore, the number of arrangements is 4! = 4 * 3 * 2 * 1, which equals 24.

Question 2: What is the probability of rolling a sum of 7 with two fair dice?

1/6 (Correct answer)
1/12
1/36
1/9

Correct answer: 1/6

When rolling two fair dice, there are 6 * 6 = 36 possible outcomes. The combinations that sum to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), totaling 6 favorable outcomes. The probability is the number of favorable outcomes divided by the total number of outcomes, which is 6/36, simplifying to 1/6.

Question 3: How many different ways can you select 3 objects from a set of 10?

120 (Correct answer)
45
30
60

Correct answer: 120

This is a combination problem because the order in which the objects are selected does not matter. The number of ways to select k objects from a set of n is given by the combination formula C(n, k) = n! / (k! * (n-k)!). For selecting 3 objects from 10, C(10, 3) = 10! / (3! * 7!) = (10 * 9 * 8) / (3 * 2 * 1) = 120.

Question 4: What is the probability of drawing a red card from a standard deck of 52 cards?

1/2 (Correct answer)
1/4
1/3
1/5

Correct answer: 1/2

A standard deck of 52 cards consists of 26 red cards (13 hearts and 13 diamonds) and 26 black cards. The probability of drawing a red card is the number of red cards divided by the total number of cards. This is 26/52, which simplifies to 1/2.

Question 5: If a fair coin is tossed 3 times, what is the probability of getting exactly 2 heads?

3/8 (Correct answer)
1/4
1/2
1/8

Correct answer: 3/8

When a fair coin is tossed 3 times, there are 2³ = 8 possible outcomes (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT). The outcomes with exactly 2 heads are HHT, HTH, and THH, which are 3 favorable outcomes. Therefore, the probability is 3 (favorable outcomes) / 8 (total outcomes) = 3/8.

Question 6: How many different ways can you arrange 3 blue balls, 2 red balls, and 1 green ball?

60 (Correct answer)
120
24
30

Correct answer: 60

This is a permutation with repetitions problem. We have a total of 6 balls (3 blue + 2 red + 1 green). The number of distinct arrangements is given by n! / (n₁! * n₂! * ... * nk!), where n is the total number of items and nᵢ is the count of each repeated item. So, 6! / (3! * 2! * 1!) = 720 / (6 * 2 * 1) = 720 / 12 = 60.

Question 7: In how many ways can you distribute 5 identical objects into 3 distinct boxes?

21 (Correct answer)
15
10
25

Correct answer: 21

This problem can be solved using the stars and bars method for distributing identical objects into distinct boxes. We have 5 identical objects (stars) and 3 distinct boxes, which requires 3-1 = 2 bars. The number of ways is given by C(n + k - 1, k - 1), where n is the number of objects and k is the number of boxes. So, C(5 + 3 - 1, 3 - 1) = C(7, 2) = 7! / (2! * 5!) = (7 * 6) / (2 * 1) = 21.

Question 8: What is the number of ways to arrange the digits 1, 2, 3, 4, and 5 in a 5-digit number?

120 (Correct answer)
60
24
30

Correct answer: 120

The number of ways to arrange n distinct digits in an n-digit number is given by n! (n factorial). Here, we have 5 distinct digits (1, 2, 3, 4, 5). Therefore, the number of arrangements is 5! = 5 * 4 * 3 * 2 * 1, which equals 120.

Question 9: If a fair die is rolled, what is the probability of rolling an even number?

1/2 (Correct answer)
1/6
1/3
1/4

Correct answer: 1/2

When a fair die is rolled, there are 6 possible outcomes: {1, 2, 3, 4, 5, 6}. The even numbers in this set are {2, 4, 6}, which are 3 favorable outcomes. The probability of rolling an even number is the number of favorable outcomes divided by the total number of outcomes, which is 3/6, simplifying to 1/2.

Free AMC 12 Combinatorics & Probability Questions and Answers

How many ways can you arrange the letters of the word 'MATH'?