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# Word Problems with the Counting Principle, Permutations, and Combinations Study Guide (page 2)

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Updated on Oct 4, 2011

## Combinations

The major difference between the combinations of a set of objects and the permutations of the same set of objects is that with combinations, the order does not matter. When the order of the objects changes, it is still considered the same combination of items.

Calculating the total number of combinations is very similar to finding the number of permutations. However, because the order does not matter, you must divide by the total number of different orders of the objects.

#### Tip:

Because the order does not matter, the total number of combinations of n objects taken n at a time is always equal to 1. For example, the total number of combinations of 3 objects when all 3 objects are used is 1. Even though you can list the objects in different orders, the set still contains the same 3 objects.

Take the next example.

How many different combinations of 3 students can be selected for a committee when there are a total of 10 students from which to choose?

Read and understand the question. This question is looking for the total number of combinations of 10 students selected 3 at a time for a committee.

Make a plan. Because the order is not important, find the number of permutations of 10 students taken 3 at a time, and then divide by the number of permutations of 3 students taken 3 at a time.

Carry out the plan. The number of permutations of 10 students taken 3 at a time is equal to 10 × 9 × 8 and the total number of permutations of 3 students taken 3 at a time is equal to 3 × 2 × 1. Divide the number of permutations of 10 students taken 3 at a time by the number of permutations of 3 students taken 3 at a time to find the total number of combinations of 10 students taken 3 at a time.

different combinations.

Check your answer. To check this solution, find the number of combinations of 10 students taken 7 at a time, which should be the same result. To do this, calculate the number of permutations of 10 students taken 7 at a time and divide this amount by the permuations of 7 students taken 7 at a time. The number of combinations is equal to

so this solution is checking.

#### Tip:

When you are looking at the total number of permutations of a set of objects, the order matters. When looking at the total number of combinations of a set of objects, the order does not matter.

Find practice problems and solutions for these concepts at Word Problems with the Counting Principle, Permutations and Combinations Practice Questions.

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