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Factorial

A factorial indicates in how many ways you can arrange n elements

n! = 1 × 2 × 3 × 4 × ··· × n

It is called n-factorial.

 


Explanation

We will play with the letters ABCD. Successively we pick more and more letters and see how many variations we can make with them. With 1 letter you only have 1! = 1 possibility

A

With 2 letters you have 2! = 1 × 2 posibilities

AB   BA

With 3 letters you have 3! = 1 × 2 × 3 = 6 possibilities

ABC   BCA   BAC
ACB   CAB   CBA

With 4 letters you already have 4! = 1 × 2 × 3 × 4 = 24 possibilities

ABCD   BACD   CABD   DABC
ABDC   BADC   CADB   DACB
ACBD   BCAD   CBAD   DBAC
ACDB   BCDA   CBDA   DBCA
ADAB   BDAC   CDAB   DCAB
ADBA   BDCA   CDBA   DCBA

That increases fast. We have seen above that

4! = 1 × 2 × 3 × 4
3! = 1 × 2 × 3
2! = 1 × 2
1! = 1

You can decide not to take any letters. That is also a possibility. So by definition zero-factorial is

0! ≝ 1

 


Additional information

The French mathematician Christian Kramp (1760 - 1826) invented the symbol n! for the factorial.

 


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