The factorial function of a number is a natural number with a factorial sign. Factorial of a number is denoted by Ị For example 4Ị = 4*3*2*1 i.e. Reducing 1 from the given number and multiplying the number till we reach 1 at last. We usually use factorials while solving permutations and combinations and in binomial expansion.  Factorial is the product of all whole numbers between 1 and n, where n should be positive. A prime number that is more or 1 less than value of a factorial is called a factorial prime number.

Factorial formula

N!=n*n-1*….*1.

Factorial table 

Applications of the factorial function 

1. Recursion- Recursively, the factorial function can be expressed as 

factorial(n) = n factorial(n – 1) .The factorial of one is just one. The factorial function is written as a recursive function in Code Example 6.27. To make it easier to refer to program addresses, we’ll suppose the program starts at 0x90.

1. Permutation –
2. Combinations 
3. Probability distribution 
4. Number theory 

Some questions related to factorial 

1.(a+2)!/a!

(a+2)!/a!

[1*2*3*…..*a*(a+1)*(a+2)]/[1*2*…..*a]

On simplifying it we get,

(a+1)(a+2)

2.(2a+2)!=2a!

[1*2*3…….(2a)*(2a+1)*(2a+2)/[1*2*3……2a]

On simplifying it we get,

(2a+1)*(2a+2)

3.(a-1)!/a(a+1)

[1*2*3…(a-1)]/[1*2*3…..(n-1)*n*(n+1)] 

On simplifying it we get,

1/[n(n+1)]

Q2. Let n denote a natural number. Show that (n+2)2 is a factor of (n+2)!+(n+1)!+n!.

Given : n – natural number 

Let (n+2)! + (n+1)! + n!….(1)

Using factorial formula,

(n+2)! = (n+2)(n+1)n!

(n+1)!=(n+1)n! ……(2) 

Substitute 2 in 1.

(n+2)(n+1)n!+(n+1)n!+n!.

 n![(n+2)(n+1)(n+1)+1]

n![(n+2)(n+1)+n+2]

take n+2 factor 

n![(n+2)((n+1)+1)]

n![(n+2)(n+2)]

n!(n+2)2 .

(n+2)2 is a factor of (n+2)(n+1)n!+(n+1)n!+n!.

Q3. 10!/10!-5!

10*9*8*7*6 = 30240. 

Q3. Evaluate the following expressions:
6!

4! × 4!

2! × 0!

5! / 0!

6! / (2! × 4!)

Ans – 6! = 1 *2 *3 * 4*5*6 = 720

4! × 4! = (1 × 2 × 3 × 4) 2 = 24 2 = 576

2! × 0! = (1 × 2 ) × 1 = 2

5! / 0! = (1 × 2 × 3 × 4× 5) / 1 = 120

6! / (2! × 4!)
= (1 × 2 × 3 × 4 × 5 × 6) / [ (1 × 2 ) × ( 1 × 2 × 3 × 4) ]
= 15

Conclusion 

The product of all the positive numbers preceding or corresponding to n is the factorial (indicated or represented as n!) for a positive number or integer (which is denoted by n) (the positive integer). There are a number of sequences in mathematics that are comparable to the factorial. Double Factorials, Multi-factorials, Super-factorials, and Hyper-factorials are only a few examples.0’s factorial equals 1’s factorial (one).