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.
Suppose we want to take a picture of three boys, Allen, Bryan and Carlos. In how many ways can the boys be arranged?
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
In how many ways can 6 people be seated in a row of 6 chairs?
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.
In how many ways can seven different books be arranged on the shelf?
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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