Question

Simplifique essa equação fatorial
$\frac{(n - r + 1)}{(n - r - 1)}$
lembrado ! nesses fatoriais ​

Answer 1

$\dfrac{(n-r+1)!}{(n-r-1)!}~=\\\\\\\\=~\dfrac{(n-r+1)~.~(n-r+1-1)~.~(n-r+1-2)!}{(n-r-1)!}\\\\\\\\=~\dfrac{(n-r+1)~.~(n-r)~.~(n-r-1)!}{(n-r-1)!}\\\\\\\\=~\dfrac{(n-r+1)~.~(n-r)~.~1}{1}\\\\\\\\=~\boxed{(n-r+1)~.~(n-r)}\\\\\\\\Podemos~ainda~aplicar~a~propriedade~distributiva,~ficando:\\\\\\=~n~.~n~-~n~.~r~-~r~.~n~-~r~.~(-r)~+~1~.~n~-~1~.~r\\\\\\\\=~n^2~-~nr~-~nr~+~r^2~+~n~-~r\\\\\\=~\boxed{n^2~-~2nr~+~n~-~r~+~r^2}$