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# Word Problems with the Counting Principle, Permutations and Combinations Practice Questions (page 2)

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#### Solutions

1. 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.
2. 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.

3. 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.
4. 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.

5. 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.
6. 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

1. How many ways can 2 books be selected out of a series of 4 books if the order is not important?
2. 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?
3. 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?

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