Factorial Identities
Key Questions

A factorial is a number multiplied by all of the integers below it.
Eg, "Four factorial"
#=4!# #=4xx3xx2xx1=24# 
Answer:
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!)(xa)+(f''(a))/(2!)(xa)^2+(f'''(a))/(3!)(xa)^3+...# Binomial Expansion
#(a+b)^n = ((n),(0))a^n + ((n),(1))a^(n1)b + ((n),(2))a^(n2)b^2 +...+((n),(n))b^n# where
#((n),(k)) = (n!)/(k!(nk)!)# 
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#