# Factorial !

The factorial function (using symbol: !) says to multiply all whole numbers from our chosen number down to 1.

So n! (read “n factorial”) means n × (n-1) × (n-2) × … × 3 × 2 × 1

## Definition of Factorial

What is the Factorial function?

The factorial function is a function represented by an exclamation mark “!”. To find the result of Factorial function, you need to multiply the number that appears in the formula with all the positives integers that smaller than it.

## The utility of the Factorial function

Usually, Factorial functions is used to calculate permutations & combinations, so you can also calculate probabilities.

Examples

If you want to hang 5 Photos on the wall, one after another you can calculate the number of possible combinations:

5! = 5×4×3×2×1 = 120 possible combinations

Other Examples:

6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362880

1! = 1

## Calculating Factorial From the Previous Value

We can easily calculate a factorial from the previous one:

 n n! n × (n-1)! Result 1 1 1 1 2 2 × 1 2 × 1! 2 3 3 × 2 × 1 3 × 2! 6 4 4 × 3 × 2 × 1 4 × 3! 24 5 5 × 4 × 3 × 2 × 1 5 × 4! 120 … … … …

So the rule is:

n! = n × (n−1)!

Which says

the factorial of any number is that number times the factorial of (that number minus 1)”

Example

7! = 7 × 6!

100! = 100 × 99!

etc.

Zero Factorial is interesting

It is generally agreed that 0! = 1.

It may seem funny that multiplying no numbers together results in 1, but let’s start from the rule:

n! = n × (n−1)!

If n = 1, then

1! = 1 × (1−1)!

1 = 1 × 0!

0! = 1/1

0! = 1

So, in many equations using 0! = 1 just makes sense.

## Table of factorial numbers

 n n! Result 0 1 1 1 1 1 2 2 × 1! 2 3 3 × 2! 6 4 4 × 3! 24 5 5 × 4! 120 6 6 × 5! 720 7 7 × 6! 5,04 8 8 × 7! 40,32 9 9 × 8! 362,88 10 10 × 9! 3,628,800 11 11 × 10! 39,916,800 12 12 × 11! 479,001,600 13 13 × 12! 6,227,020,800 14 14 × 13! 87,178,291,200 15 15 × 14! 1,307,674,368,000 16 16 × 15! 20,922,789,888,000 17 17 × 16! 355,687,428,096,000 18 18 × 17! 6,402,373,705,728,000 19 19 × 18! 121,645,100,408,832,000 20 20 × 19! 2,432,902,008,176,640,000 21 21 × 20! 51,090,942,171,709,440,000 22 22 × 21! 1,124,000,727,777,607,680,000 23 23 × 22! 25,852,016,738,884,976,640,000 24 24 × 23! 620,448,401,733,239,439,360,000 25 25 × 24! 15,511,210,043,330,985,984,000,000

## History of Factorial

Factorial was used by Indian scholars to calculate the permutations at least as early as the 12th century.

Then, Fabian Stedman described factorials as applied to change ringing, a musical art involving the ringing of many tuned bells In 1677. Stedman gives a statement of a factorial:

“ Now the nature of these methods is such, that the changes on one number comprehends [includes] the changes on all lesser numbers … in so much that a compleat Peal of changes on one number seemeth to be formed by uniting of the compleat Peals on all lesser numbers into one entire body”

In 1808, French mathematician Christian Kramp introduced The notation n!.