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What Is the Factorial?

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In mathematics symbols that have certain meanings in the English language can take on very specialized and different meanings. Perhaps this is most apparent when looking at an expression such as 3! No, we did not use the exclamation point to show that we’re excited about three, and we shouldn’t read the last sentence with emphasis. In mathematics the expression 3! is read as "three factorial" and is really a shorthand way to denote the multiplication of several consecutive whole numbers.

Since there are many places throughout mathematics and statistics where we need to multiply numbers together, the factorial is quite useful. Some of the main places where it shows up are combinatorics and calculus.

Definition

The definition of the factorial is that for any positive whole number n, the factorial:

n! = n x (n -1) x (n - 2) x . . . x 2 x 1

Examples for Small Values

First we will look at a few examples of the factorial with small values of n:

  • 1! = 1
  • 2! = 2 x 1 = 2
  • 3! = 3 x 2 x 1 = 6
  • 4! = 4 x 3 x 2 x 1 = 24
  • 5! = 5 x 4 x 3 x 2 x 1 = 120
  • 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
  • 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
  • 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320
  • 9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362880
  • 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3628800

As we can see the factorial gets very large very quickly. Something that may seem small, such as 20! actually has 19 digits.

Factorials are easy to compute, but they can be somewhat tedious to calculate. Fortunately many calculators have a factorial key (look for the ! symbol). This function of the calculator will automate the multiplications.

A Special Case

One other value of the factorial, and one for which the standard definition above does not hold, is that of zero factorial. If we follow the formula, then we would not arrive at any value for 0!. There are no positive whole numbers less than 0. For several reasons, it is appropriate to define 0! = 1. The factorial for this value shows up particularly in the formulas for combinations and permutations.

More Advanced Calculations

When dealing with calculations, it is important to think before we press the factorial key on our calculator. To calculate an expression such as 100!/98! there are a couple of different ways of going about this.

One way is to use a calculator to find both 100! and 98!, then divide one by the other. Although this is a direct way to calculate, it has some difficulties associated with it. Some calculators cannot handle expressions as large as 100! = 9.33262154 x 10157. (The expression 10157 is scientific notation that means that we multiply by 1 followed by 157 zeros.) Not only is this number massive, but it is also only an estimate to the real value of 100!

Another way to simplify an expression with factorials like the one seen here does not require a calculator at all. The way to approach this problem is to recognize that we can rewrite 100! not as 100 x 99 x 98 x 97 x . . . x 2 x 1, but instead as 100 x 99 x 98! The expression 100!/98! now becomes (100 x 99 x 98!)/98! = 100 x 99 = 9900.

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