Multiplication is certainly known to everyone and so are factorial notations. Factorial operation is a symbol used to represent multiplication operation.
Recall the pages of mathematics text books which stated that the factorial of any positive number n is recognized as the product of n and the entire positive integers that are less than n. this could be written as n! = n(n – 1)(n – 2) .. (3) (2) (1). While dealing with combinations and permutations, you would be required to know better about factorial.
Factorial n – Definition
A factorial is known to be represented by the (!) sign. When we come across n! Which is also known as the “n factorial”; we would come up with the fact that a factorial is known to be the product of all the whole numbers that lie between 1 and n, where n is supposed to be positive.
For instance, consider:
1! = 1 = 1
2! = 2 x 1 = 2
3! = 3 x 2 x 1 = 6
4! = 4 x 3 x 2 x 1 = 24
Also, know that 0! is regarded as a special case factorial.
0! = 1
It is considered to be special because there aren’t any positive numbers less than zero. It is also important to recall that we had earlier defined a factorial as the product of numbers between 1 and n. When we say that 0! = 1, we are claiming that there are no numbers whose product is 1. The mathematics and reasoning involved with it could be difficult and complicated. This statement therefore needs to be accepted without actually getting any explanations for the same.
Thactsually turns out to be true mathematically, and allows us to redefine n! as below:
n! = n x (n – 1)!
1! = 1 x 0! = 1
2! = 2 x 1! = 2
5! = 5 x 4! = 120
The above explanations allow us to manipulate and break down factorials. Some more useful properties of the factorials are:
- n! = n(n – 1)!
- (n x m)! ≠ n! x m!
- (n + m)! ≠ n! + m!
- (n – m)! ≠ n! – m!
The last two properties are probably the most important ones to remember. The factorial sign doesn’t get distributed across operations like addition and subtraction.
The explanation here would help you understand “n factorial” better.