Factorial !

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

Factorial Symbol

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

We 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: 9!=362,880. Work out the value of 10!

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)", so 10! = 10 × 9!, ... and 125! = 125 × 124!, etc.

What About "0!"

Zero Factorial is interesting ... it is generally agreed that 0! = 1.

It may seem funny that in this case multiplying no numbers together results in 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

What About Decimals?

Can we have factorials for numbers like 0.5 or -3.217?

Yes we can! But we 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?