The factorial function is defined as:

We defineF(n) =n(n- 1)(n- 2)(n- 3)^{ ... }(2)(1)

wherenis a non-negative integer.

The factorial function
*F*(*n*)
is also represented as "
*n*!
", read "
*n*
factorial."

*Examples.*

5! | = | 5(4)(3)(2)(1) = 120 | |

3! | = | 3(2)(1) = 6 | |

10! | = | 10(9)(8)(7)(6)(5)(4)(3)(2)(1) = 3628800 | |

0! | = | 1;;(by definition) | |

6! | = | 6(5)(4)(3)(2)(1) = 720 |

*Example 1*: If 6 children must form a line, in how many ways can they
arrange themselves?

6 different children can stand in the first spot, 5 in the second spot (since 1 is already occupying the first spot), 4 in the third spot (since 2 are already occupying spots), etc.

Thus, they can arrange themselves in 6(5)(4)(3)(2)(1) = 6! = 720 different ways.

*n*!
gives the number of ways
*n*
distinct items can be arranged in an order.

*Example 2*: In how many ways can the letters of the word TRIANGLE be
arranged?

8! = 40320 different ways.

Occasionally, we will encounter a situation in which the choices are not
distinct. For example, in how many ways can he letters of the word ALGEBRA be
arranged?

Since an arrangement with the first A in the
5^{th}
spot and the last A in the
6^{th}
spot is no different than an arrangement with the first A in the
6^{th}
spot and the last A in the
5^{th}
spot, we must account for the
overlap. The total number of possibilities is
= = 2520
. We divide by
2!
because
*n*!
is the number of ways
*n*
A's can be
arranged.

To find the total number of possibilities when choices are not distinct, divide by the factorial of the number of choices that are the same. If 2 choices are the same as each other, and 2 different choices are the same as each other, divide by 2!2!. If 2 choices are the same as each other, and 3 different choices are the same as each other, divide by 2!3!.

*Example 3*: In how many ways can the letters of the word BANANA be
arranged?

There are 6 letters, 3 A's, and 2 N's. Thus, the letters can be arranged in
= = 60
different ways.