Factorial !
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The factorial function (symbol: !) just 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
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"4!" is usually pronounced "4 factorial". 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 |
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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: the number times the factorial of (1 smaller than the 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 |
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= 7 × 6 × 5 = 210 |
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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 |
51,090,942,171,709,400,000 |
| 22 |
1,124,000,727,777,610,000,000 |
| 23 |
25,852,016,738,885,000,000,000 |
| 24 |
620,448,401,733,239,000,000,000 |
| 25 |
15,511,210,043,331,000,000,000,000 |
As you can see, it gets big quickly!
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 |
And it still follows the rule that "the factorial of any number is: the number times the factorial of (1 smaller than the number)", because
(3/2)! = (3/2) × (1/2)!
(5/2)! = (5/2) × (3/2)!
Can you figure out what (7/2)! is?
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