Word Problems with the Counting Principle, Permutations and Combinations Practice Questions

To review these concepts, go to Word Problems with the Counting Principle, Permutations, and Combinations Study Guide.

Word Problems with the Counting Principle, Permutations and Combinations Practice Questions

Practice 1

Problems

Charlotte is picking out her class ring. She can select from a ruby, an emerald, or an opal stone, and she can also select silver or gold for the metal. How many different combinations of one stone and one type of metal can she choose?
Pete has 5 different ties that match with 3 different shirts. How many shirt-and-tie combinations can he make if he selects one shirt and one tie?
How many different sandwiches can be made with 2 choices of bread, 3 choices of toppings, and 2 choices of meat if one choice is selected from each category?

Solutions

Read and understand the question. This question is looking for the total number of possibilities when there are 3 different choices of stone and 2 choices of metal for a ring.

Make a plan. Take the total number of choices for each category and multiply them.

Carry out the plan. There are 3 different choices of stone, and two different choices of metal. Multiply these choices: 3 × 2 = 6 different ways to design the ring.

Check your answer. To check this answer, divide the total number of outcomes by the number of choices in the first category. This becomes 6 ÷ 3 = 2, which is the number of choices of metal. This answer is checking.

Read and understand the question. This question is looking for the total number of possibilities when there are 5 different choices of ties and 3 different choices of shirts.

Make a plan. Take the total number of choices for each category and multiply them.

Carry out the plan. There are 5 different choices of ties, and 3 different choices of shirts. Multiply these choices: 5 × 3 = 15 different shirt-and-tie combinations.

Check your answer. To check this answer, divide the total number of outcomes by the number of choices for a tie. This becomes 15 ÷ 5 = 3, which is the number of choices of a shirt. This answer is checking.

Read and understand the question. This question is looking for the total number of possibilities when making a sandwich from 2 choices of bread, 3 choices of toppings, and 2 choices of meat. One selection from each category will be chosen.

Make a plan. Take the total number of choices for each category and multiply them.

Carry out the plan. There are 2 choices of bread, 3 choices of toppings, and 2 choices of meat. Multiply these choices: 2 × 3 × 2 = 12 different ways.

Check your answer. To check this answer, divide the total number of outcomes by the number of choices in the first category. This becomes 12 ÷ 2 = 6. Next, divide this amount by the number of choices in the second category. This becomes 6 ÷ 3 = 2, which is the number of choices in the third category. This answer is checking.

Practice 2

Problems

How many different permutations are there of 4 objects taken 4 at a time?
There are 6 trophies on a shelf. How many different orders of all 6 trophies can be made?
How many ways can 7 students come in first, second, and third place in a geography contest, if only 1 student can earn each place?

Solutions

Read and understand the question. This question is looking for the total number of permutations, or orders, of 4 different objects taken 4 at a time.

Make a plan. Multiply the number of choices for each placement of the objects. An object can be used only once.

Carry out the plan. There are 4 choices for the first object, 3 for the second, 2 for the third, and 1 for the fourth. The number of permutations is therefore 4 × 3 × 2 × 1 = 24.

Check your answer. One way to check this solution is to divide the total number of permutations by each of the factors that were multiplied to see if the result is 1: 24 ÷ 4 = 6, 6 ÷ 3 = 2, and 2 ÷ 2 = 1, so this solution is checking.

Read and understand the question. This question is looking for the total number of permutations, or orders, of 6 different trophies taken 6 at a time.

Make a plan. Multiply the number of choices for each placement of the objects. An object can be used only once.

Carry out the plan. There are 6 choices for the first object, 5 for the second, 4 for the third, 3 for the fourth, 2 for the fifth, and only 1 for the sixth.

The number of permutations is therefore 6× 5 × 4 × 3 × 2 × 1 = 720.

Check your answer. One way to check this solution is to divide the total number of permutations by each of the factors that were multiplied to see if the result is 1.

720 ÷ 6 = 120

120 ÷ 5 = 24

24 ÷ 4 = 6

6 ÷ 3 = 2

and

2 ÷ 2 = 1

so this solution is checking.

Read and understand the question. This question is looking for the total number of permutations, or orders, of 7 different objects taken 3 at a time.

Make a plan. Multiply the number of choices for each placement of the objects until 3 factors are used. An object can be used only once.

Carry out the plan. There are 7 choices for the first object, 6 for the second, and 5 for the third. The number of permutations is therefore 7 × 6 × 5 = 210.

Check your answer. One way to check this solution is to divide the total number of permutations by each of the factors that were multiplied to see if the result is 1.

210 ÷ 7 = 30

30 ÷ 6 = 5

and

5 ÷ 5 = 1

so this solution is checking.

Practice 3

Problems

How many ways can 2 books be selected out of a series of 4 books if the order is not important?
Tyler and his family are going on a trip. He would like to select some movies to watch in the car while traveling. How many different combinations of movies are there if he selects 4 movies to watch out of a total of 8?
There are 6 students in a club. For a certain activity, 4 of the students need to form a separate group. How many ways can this group of 4 students be formed?

Solutions

Read and understand the question. This question is looking for the total number of combinations of 2 books selected from a series of 4 books.

Make a plan. Because the order is not important, find the number of permutations of 4 books taken 2 at a time, and then divide by the number of permutations of 2 books taken 2 at a time.

Carry out the plan. The number of permutations of 4 students taken 2 at a time is equal to 4 × 3 and the total number of permutations of 2 books taken 2 at a time is equal to 2 × 1. Divide to find the total number of combinations of 4 books taken 2 at a time: different combinations.

Check your answer. To check this solution, reevaluate the number of combinations. The number of combinations is equal to, so this solution is checking.

Read and understand the question. This question is looking for the total number of combinations of 8 movies taken 4 at a time.

Make a plan. Because the order is not important, find the number of permutations of 8 movies taken 4 at a time, and then divide by the number of permutations of 4 movies taken 4 at a time.

Carry out the plan. The number of permutations of 8 movies taken 4 at a time is equal to 8 × 7 × 6 × 5 and the total number of permutations of 4 movies taken 4 at a time is equal to 4 × 3 × 2 × 1. Divide to find the total number of combinations of 8 movies taken 4 at a time:
different combinations.

Check your answer. To check this solution, reevaluate the number of combinations. The number of combinations is equal to
so this solution is checking.
Read and understand the question. This question is looking for the total number of combinations of 6 students taken 4 at a time.

Make a plan. Because the order is not important, find the number of permutations of 6 students taken 4 at a time, and then divide by the number of permutations of 4 students taken 4 at a time.

Carry out the plan. The number of permutations of 6 students taken 4 at a time is equal to 6 × 5 × 4 × 3 and the total number of permutations of 4 students taken 4 at a time is equal to 4 × 3 × 2 × 1. Divide to find the total number of combinations of 6 students taken 4 at a time:

different combinations.

Check your answer. To check this solution, reevaluate the number of combinations. The number of combinations is equal to

so this solution is checking.

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Word Problems with the Counting Principle, Permutations and Combinations Practice Questions