Combination and permutation word problems involve the number of ways of selecting r items out of n items, where repetitions are not allowed. If the order is important, meaning changing the order changes the result, we want to use the formula for permutations. If the order is not important, meaning changing the order does not change the result, we want to use the formula for combinations.

Test Objectives
• Demonstrate the ability to solve a word problem that involves a permutation
• Demonstrate the ability to solve a word problem that involves a combination
Counting Theory Practice Test:

#1:

Instructions: Solve each word problem.

a) A group of 30 people are running in a marathon. The first 5 people to finish the race will advance to the finals.

b) A group of 16 people need to take an elevator to the penthouse. Due to the current fire code, they can only ride in groups of 8. As a group, they will need to decide who will take the elevator on the second trip up.

#2:

Instructions: Solve each word problem.

a) A softball team, which consists of a total of 17 players needs to choose 1 captain and 1 co-captain.

b) A group of 35 juniors are going to run a race. The top 3 runners will earn gold, silver, and bronze medals.

#3:

Instructions: Solve each word problem.

a) James has 7 total classes, each with a homework assignment that is due. He only has time to do 4 of the 7.

b) Jennifer has been tasked with setting the combination in a five-digit lock. She wants to use the numbers 1, 0, 9, 6, 2 and doesn’t care what order they are in.

#4:

Instructions: Solve each word problem.

a) Brenda and Jessica are planning trips to 2 countries this year. There are a total of 5 countries that they would like to visit. They are deciding which countries to skip.

b) A soccer team consists of 15 total players. They must choose 2 players to refill the water coolers.

#5:

Instructions: Solve each word problem.

a) At XYZ Corporation, there are 2 jobs available, computer programmer and customer experience. There are 15 total applicants for the 2 jobs.

b) At ZYD Corporation, there are 50 applicants for 2 manager positions.

Written Solutions:

#1:

Solutions:

a) Combination; 142,506

b) Combination; 12,870

#2:

Solutions:

a) Permutation; 272

b) Permutation; 39,270

#3:

Solutions:

a) Combination; 35

b) Permutation; 120

#4:

Solutions:

a) Combination; 10

b) Combination; 105

#5:

Solutions:

a) Permutation; 210

b) Combination; 1,225