Question

Me ajudem a responder duas questões de análise combinatória que eu preciso entregar amanhã, por favor?!!

a) (n - 3)! / (n - 5)!
b) (n + 2)! / (n - 1)!

Answer 1

a)

$\dfrac{(n-3)!}{(n-5)!}~=\\\\\\\\=~\dfrac{(n-3)~.~(n-3-1)~.~(n-3-1-1)!}{(n-5)!}\\\\\\\\=~\dfrac{(n-3)~.~(n-4)~.~(n-5)!}{(n-5)!}\\\\\\\\=~\dfrac{(n-3)~.~(n-4)~.~1}{1}\\\\\\\\=~\boxed{(n-3).(n-4)}\\\\\\ou,~aplicando~a~propriedade~distributiva~da~multiplicacao\\\\\\=~\boxed{n^2-7n+12}$

b)

$\dfrac{(n+2)!}{(n-1)!}~=\\\\\\\\=~\dfrac{(n+2)~.~(n+2-1)~.~(n+2-1-1)~.~(n+2-1-1-1)!}{(n-1)!}\\\\\\\\=~\dfrac{(n+2)~.~(n+1)~.~(n)~.~(n-1)!}{(n-1)!}\\\\\\\\=~\dfrac{(n+2)~.~(n+1)~.~n~.~1}{1}\\\\\\\\=~\boxed{(n+2).(n+1).n}\\\\\\ou,~aplicando~a~propriedade~distributiva~da~multiplicacao\\\\\\=~\boxed{n^3+3n^2+2n}$