# Permutations P(n,n)

In these lessons, we will look at the number of permutations of n things taken n at a time.

A permutation is an arrangement, or listing, of objects in which the order is important.

Here, we will look at examples of the number of permutations of n things taken n at a time. In subsequent lessons, we will consider the number of permutations of n things taken r at a time.

Example:
Suppose we want to take a picture of three boys, Allen, Bryan and Carlos. In how many ways can the boys be arranged?

Solution:
They can be arranged in any of several ways.

Allen Bryan Carlos
Allen Carlos Bryan
Bryan Allen Carlos
Bryan Carlos Allen
Carlos Allen Bryan
Carlos Bryan Allen

There are 3 choices for the first boy, 2 choices for the second and 1 choice for the third, so the total number of permutations is 3 x 2 x 1 = 6. The 3 boys can be arranged in 6 ways.

In this example, the symbol P(3, 3) represents the number of permutations of 3 things taken 3 at a time.

P(3, 3) = 3 × 2 × 1 = 6

Example:
In how many ways can 6 people be seated in a row of 6 chairs?

Solution:
We can use the permutation formula P(6, 6) which is 6 things taken 6 at a time.

P(6, 6) = 6 × 5 × 4 × 3 × 2 × 1

Six people can be seated in 720 ways.

Example:
In how many ways can seven different books be arranged on the shelf?

Solution:
We can use the permutation formula P(7, 7) which is 6 things taken 6 at a time.

P(7, 7) = 7 × 6 × 5 × 4 × 3 × 2 × 1

The books can be arranged in 5,040 ways.

The following video gives another example of the permutation problem.

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