# Factorial Identities

## Key Questions

• A factorial is a number multiplied by all of the integers below it.

Eg, "Four factorial" =4!$= 4 \times 3 \times 2 \times 1 = 24$

Many things in various areas of mathematics.

#### Explanation:

Here are a few examples:

Probability (Combinatorics)

If a fair coin is tossed 10 times, what is the probability of exactly $6$ heads?

Answer: (10!)/(6! 4! 2^10)

Series for sin, cos and exponential functions

sin(x) = x - x^3/(3!) + x^5/(5!) -x^7/(7!)+...

cos(x) = 1 - x^2/(2!) + x^4/(4!) - x^6/(6!) +...

e^x = 1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) + ...

Taylor Series

f(x) = f(a)/(0!)+(f'(a))/(1!)(x-a)+(f''(a))/(2!)(x-a)^2+(f'''(a))/(3!)(x-a)^3+...

Binomial Expansion

${\left(a + b\right)}^{n} = \left(\begin{matrix}n \\ 0\end{matrix}\right) {a}^{n} + \left(\begin{matrix}n \\ 1\end{matrix}\right) {a}^{n - 1} b + \left(\begin{matrix}n \\ 2\end{matrix}\right) {a}^{n - 2} {b}^{2} + \ldots + \left(\begin{matrix}n \\ n\end{matrix}\right) {b}^{n}$

where ((n),(k)) = (n!)/(k!(n-k)!)

• The factorial of a number n is denoted as n!

This is the product of all numbers from 1 to n.

So we have n! = n * (n - 1) * (n -2) * … * 2 * 1

So let us take 6!

6! = 6 * 5 * 4 * 3 * 2 * 1

6! = 720