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.*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.*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 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.

*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.

*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?

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