Question

What are factorials used for?

Answer 1

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

`(a+b)^n = ((n),(0))a^n + ((n),(1))a^(n-1)b + ((n),(2))a^(n-2)b^2 +...+((n),(n))b^n`

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