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

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Updated on Oct 4, 2011

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

  1. 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?
  2. 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?
  3. 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

  1. 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.
  2. 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.

  3. 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.
  4. 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.

  5. 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.
  6. 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

  1. How many different permutations are there of 4 objects taken 4 at a time?
  2. There are 6 trophies on a shelf. How many different orders of all 6 trophies can be made?
  3. 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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