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Home : Math & Science : Math Study Guides : Algebra II : Discrete Functions : The Factorial Function
The Factorial Function
The Factorial Function
The factorial function is defined as:
F(n) = n(n - 1)(n - 2)(n - 3) ... (2)(1)
We define F(0) = 1 and F(1) = 1.
where n is a non-negative integer.
The factorial function F(n) is also represented as "n!", read "n
factorial."
Examples.
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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