What are factorials used for?

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Answer 1 · 2015-09-06T22:40:28

Many things in various areas of mathematics.

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