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:
|
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 1st, 2nd and 3rd?
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 1st, 2nd and 3rd.
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 |
As you can see, it gets big quickly.
If you need more, try the Full Precision Calculator.
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
