In this lesson, we will explore the concept of a permutation. A permutation is an arrangement or selection in which the order matters.

Consider this scenario: a college student has just purchased a new bookshelf for her dorm room and she has a total of n books that she wishes to arrange on the shelf. How many different ways can she arrange her books?

For the first spot on the shelf, she has n different options for the placement of a book. Next, since one book has already been placed, she now has n-1 books to place in the second spot. She will then have n-2 books for the third spot, n-3 books for the fourth spot, and so on. Suppose she has nine books in total, the arrangement of the bookshelf will look like this:

$\underline{9} \ \underline{8} \ \underline{7} \ \underline{6}\ \ldots \ \underline{1}$

So there are:

$9 \times 8 \times 7 \times \ldots \times 1 = 362,880$

ways to arrange the books.

For a general arrangement of n items, we have:

$n(n-1)(n-2) \ldots 1 = n!$

Here, the ! is the factorial operator.

Arranging n items without repetition results in n! a number of permutations.

Note the following identities:

• 0! = 1
• The factorial operation is undefined for numbers less than zero.