# Factorials with variables?

Can someone give me a good explination of how to work with factorials with variables for example

to simplify (n+1)!/n! you get (n+1) as an answer, but how?

and how would I go about doing one like (2n+2)!/2n!

Notice that:
2! = 2*(1) = 2*1!
3! = 3*(2*1) = 3*2!
4! = 4*(3*2*1) = 4*3!

n! = n*(n - 1)!
(n + 1)! = (n + 1)*n!

A factorial can be written as the first factor times the factorial of the next factor, ie (n + 1)! = (n + 1)*n!. Applying this formula, we obtain:
(n + 1)!/n!
= (n + 1)*n!/n!
= n + 1

For (2n + 2)!/(2n)!, just apply the formula twice:
(2n + 2)!/(2n)!
= (2n + 2)*(2n + 1)!/(2n)!
= (2n + 2)*(2n + 1)*(2n)!/(2n)!
= (2n + 2)*(2n + 1)

I hope this helps!
answered Mar 6, 2011 by anonymous
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