Identitate

[Marius Stănean]

$n\ge m\ge r\ge p\Longrightarrow \dbinom{n}{m}\dbinom{m}{r}\dbinom{r}{p}=\dbinom{n}{p}\dbinom{n-p}{r-p}\dbinom{n-r}{m-r}$

[dangerous storm]

Aceasta identitate este evident adevarata dupa ce scriem fiecare binom.Ar trebui sa demonstram ca $\frac{n!}{m!(n-m)!}\cdot\frac{m!}{r!(m-r)!}\cdot\frac{r!}{p!(r-p)!}=\frac{n!}{p!(n-p)!}\cdot\frac{(n-p)!}{(r-p)!(n-r)!}\cdot\frac{(n-r)!}{(m-r)!(n-m)!}$,ceea ce este echivalent cu $\frac{1}{(m-r)!}=\frac{1}{(m-r)!}$,aceasta ultima egalitate fiind evident adevarata.