Question

simplificar:
a) n! (n-1)!
________
(n-2)! (n+1)!

b) (2n)!
_____
n!

Answer 1

Usando a definição de fatorial :
n! = n * (n - 1) * (n - 2) * .... * 3 * 2 * 1
e que 
n! = n * (n - 1) * (n - 2) * ... * (k + 1) * (n - k)! 
Logo:
a) $\mathsf{ \dfrac{n!(n - 1)!}{(n - 2)!(n+1)!} = \dfrac{n!(n-1)!}{(n-2)!(n! * (n+1))} = \dfrac{(n-2)!(n-1)}{(n-2)!(n+1)} = \dfrac{(n-1)}{(n+1)} }$

b)
$\mathsf{ \dfrac{(2n)!}{n!} = \dfrac{((2n)(2n-1)(2n-2)...(n+1)) \times n! }{n!} = } \\\\ \mathsf{(2n)(2n-1)(2n-2)...(n+1)}$