Verificare l’uguaglianza $\displaystyle \binom{k}{2} + \binom{n-k}{2} +k(n-k)- \binom{n}{n-2}=0$

SOLUZIONE

$\displaystyle \frac{k!}{2!\cdot (k-2)!}+\frac{(n-k)!}{2!\cdot (n-k-2)!}+kn -k^2-\frac{n!}{(n-2)!\cdot 2!}=$

$=\displaystyle \frac{k(k-1)}{2}+\frac{(n-k)(n-k-1)}{2}+kn -k^2-\frac{n(n-1)}{ 2}=$

$=\displaystyle \frac{k^2-k}{2}+\frac{n^2-nk-n-nk+k^2+k}{2}+kn -k^2-\frac{n^2-n}{ 2}=$

$=\displaystyle k^2-k+n^2-nk-n-nk+k^2+k+2kn -2k^2-n^2+n=0$.