Factorial Function ! (2024)

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

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

Examples:

4! = 4 × 3 × 2 × 1 = 24
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
1! = 1

We usually say (for example) 4! as "4 factorial", but some people say "4 shriek" or "4 bang"

Calculating From the Previous Value

We can easily calculate a factorial from the previous one:

As a table:

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

To work out 6!, multiply 120 by 6 to get 720
To work out 7!, multiply 720 by 7 to get 5040
And so on

Example: 9! equals 362,880. Try to calculate 10!

10! = 10 × 9!

10! = 10 × 362,880 = 3,628,800

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)"

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 multiplying no numbers together results in 1, but let's follow the pattern backwards from, say, 4! like this:

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

Example: how many ways can we arrange letters (without repeating)?

For 1 letter "a" there is only 1 way: a
For 2 letters "ab" there are 1×2=2 ways: ab, ba
For 3 letters "abc" there are 1×2×3=6 ways: abc acb cab bac bca cba
For 4 letters "abcd" there are 1×2×3×4=24 ways: (try it yourself!)
etc

The formula is simply n!

Now ... how many ways can we arrange no letters? Just one way, an empty space:

So 0! = 1

Where is Factorial Used?

One area they are used is in Combinations and Permutations. We had an example above, and here is a slightly different example:

Example: How many different ways can 7 people come 1^st, 2^nd and 3^rd?

The list is quite long, if the 7 people are called a,b,c,d,e,f and g then the list includes:

abc, abd, abe, abf, abg, acb, acd, ace, acf, ... etc.

The formula is 7!(7−3)! = 7!4!

Let us write the multiplies out in full:

7 × 6 × 5 × 4 × 3 × 2 × 14 × 3 × 2 × 1 = 7 × 6 × 5

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

7 × 6 × 5 = 210

So there are 210 different ways that 7 people could come 1^st, 2^nd and 3^rd.

Done!

Example: What is 100! / 98!

Using our knowledge from the previous example we can jump straight to this:

100!98! = 100 × 99 = 9900

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

Interesting Facts

Six weeks is exactly 10! seconds (=3,628,800)

Here is why:

Seconds in 6 weeks:		60 × 60 × 24 × 7 × 6
Factor some numbers:		(2 × 3 × 10) × (3 × 4 × 5) × (8 × 3) × 7 × 6
Rearrange:		2 × 3 × 4 × 5 × 6 × 7 × 8 × 3 × 3 × 10
Lastly 3×3=9:		2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10

There are 52! ways to shuffle a deck of cards.

That is 8.0658175... × 10⁶⁷

Just shuffle a deckof cards and it is likely that you are the first person ever with that particular order.

There are about 60! atoms in the observable Universe.

60! is about 8.320987... × 10⁸¹ and the current estimates are between 10⁷⁸ to 10⁸² atoms in the observable Universe.

70! is approximately 1.197857... 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

A Close Formula!

n! ≈ (ne)ⁿ √2πn

The "≈" means "approximately equal to". Let us see how good it is:

n	n!	Close Formula (to 2 Decimals)	Accuracy (to 4 Decimals)
1	1	0.92	0.9221
2	2	1.92	0.9595
3	6	5.84	0.9727
4	24	23.51	0.9794
5	120	118.02	0.9835
6	720	710.08	0.9862
7	5040	4980.40	0.9882
8	40320	39902.40	0.9896
9	362880	359536.87	0.9908
10	3628800	3598695.62	0.9917
11	39916800	39615625.05	0.9925
12	479001600	475687486.47	0.9931

If you don't need perfect accuracy this may be useful.

Note: it is called "Stirling's approximation" and is based on a simplifed version of the Gamma Function.

What About Negatives?

Can we have factorials for negative numbers?

Yes ... but not for negative integers.

Negative integer factorials (like -1!, -2!, etc) are undefined.

Let's start with 3! = 3 × 2 × 1 = 6 and go down:

2!	=	3! / 3	=	6 / 3	=	2
1!	=	2! / 2	=	2 / 2	=	1
0!	=	1! / 1	=	1 / 1	=	1	which is why 0!=1
(−1)!	=	0! / 0	=	1 / 0	=	?	oops, dividing by zero is undefined

And from here on down all integer factorials are undefined.

What About Decimals?

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

Yes we can! But we need to use the Gamma Function (advanced topic).

Factorials can also be negative (except for negative integers).

Half Factorial

But I can tell you the factorial of half (½) is half of the square root of pi .

Here are some "half-integer" factorials:

(−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?

Double Factorial!!

A double factorial is like a normal factorial but we skip every second number:

8!! = 8 × 6 × 4 × 2 = 384
9!! = 9 × 7 × 5 × 3 × 1 = 945

Notice how we multiply all even, or all odd, numbers.

Note: if we want to apply factorial twice we write (n!)!

2229, 2230, 7006, 2231, 7007, 9080, 9081, 9082, 9083, 9084

Combinations and Permutations Gamma Function Numbers Index