Number System: Factorial Function

Factorial !

Example: 4! is shorthand for 4 x 3 x 2 x 1

The factorial function (symbol: !) means to multiply a series of descending natural numbers. Examples: 4! = 4 × 3 × 2 × 1 = 24 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040 1! = 1

4! is usually pronounced "4 factorial", but some people even say "4 shriek" or "4 bang"

Calculating From the Previous Value

You can easily calculate a factorial from the previous one:

n	n!
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
6	etc	etc

Example: What is 10! if you know that 9!=362,880 ?

10! = 10 × 9!
10! = 10 × 362,880 = 3,628,800

So the rule is:

n! = n × (n-1)!

Which just says "the factorial of any number is that number times the factorial of (1 smaller than that number)", hence 10! = 10 × 9!, or even 125! = 125 × 124!

What About "0!"

Zero Factorial is interesting ... it is generally agreed that 0! = 1.
It may seem funny that multiplying no numbers together gets you 1, but it helps simplify a lot of equations.

Where is Factorial Used?

Factorials are used in many areas of mathematics, but particularly in Combinations and Permutations

Example: What is 7! / 4!

Let us write them out in full:

7 × 6 × 5 × 4 × 3 × 2 × 1		= 7 × 6 × 5 = 210
4 × 3 × 2 × 1

That was neat. The 4 × 3 × 2 × 1 "cancelled out", leaving only 7 × 6 × 5

A Small List

n	n!
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5,040
8	40,320
9	362,880
10	3,628,800
11	39,916,800
12	479,001,600
13	6,227,020,800
14	87,178,291,200
15	1,307,674,368,000
16	20,922,789,888,000
17	355,687,428,096,000
18	6,402,373,705,728,000
19	121,645,100,408,832,000
20	2,432,902,008,176,640,000
21	51,090,942,171,709,440,000
22	1,124,000,727,777,607,680,000
23	25,852,016,738,884,976,640,000
24	620,448,401,733,239,439,360,000
25	15,511,210,043,330,985,984,000,000

As you can see, it gets big quickly!
If you need more, try our Full Precision Calculator.

Some Bigger Values

70! is approximately 1.1978571669969891796072783721 x 10¹⁰⁰, which is just larger than a Googol (the digit 1 followed by one hundred zeros).
100! is approximately 9.3326215443944152681699238856 x 10¹⁵⁷
200! is approximately 7.8865786736479050355236321393 x 10³⁷⁴

... Advanced Topic Follows ...

What About Decimals?

Can you have factorials for numbers like 0.5 or -3.217?
Yes you can! But you need to get into a subject called the "Gamma Function", which is beyond this simple page.

Half Factorial

But I can tell you the factorial of half (½) is half of the square root of pi = (½)√π, and so some "half-integer" factorials are:

n	n!
(-½)!	√π
(½)!	(½)√π
(3/2)!	(3/4)√π
(5/2)!	(15/8)√π

It still follows the rule that "the factorial of any number is that number times the factorial of (1 smaller than that number)", because

(3/2)! = (3/2) × (1/2)!
(5/2)! = (5/2) × (3/2)!

Can you figure out what (7/2)! is?

Posted in: Factorial Function, Number System