FREE AMC 12 Combinatorics & Probability Questions and Answers

Question 1

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

Accepted Answer

24

Answer

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?

Accepted Answer

1/6

Answer

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?

Accepted Answer

120

Answer

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?

Accepted Answer

1/2

Answer

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?

Accepted Answer

3/8

Answer

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.

AMC 12 - - American Mathematics Contest 12 Practice Test

AMC 12 - - American Mathematics Contest 12 Practice Test

FREE AMC 12 Combinatorics & Probability Questions and Answers