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Discrete Functions

The Factorial Function

Terms

Problems

The Factorial Function

The factorial function is defined as:

F(n) = n(n - 1)(n - 2)(n - 3) ... (2)(1)
where n is a non-negative integer.
We define F(0) = 1 and F(1) = 1 .

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 5th spot and the last A in the 6th spot is no different than an arrangement with the first A in the 6th spot and the last A in the 5th 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.

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