# Math Help - Prove this identity

1. ## Prove this identity

For positive integers $r$, define $x^{\underline{r}}= x(x-1) \cdots (x-r+1)$

Show that $x^n = \Big\{^n_1 \Big\}x^{\underline{1}} + \Big\{^n_2 \Big\}x^{\underline{2}} + \cdots + \Big\{^n_n \Big\}x^{\underline{n}}$

It looks so similar to $n2^{n-1} = 1\binom{n}{1}+2\binom{n}{2}+ \cdots + n\binom{n}{n}$ but I have no idea how to prove it

Any ideas?

2. Originally Posted by usagi_killer
For positive integers $r$, define $x^{\underline{r}}= x(x-1) \cdots (x-r+1)$

Show that $x^n = \Big\{^n_1 \Big\}x^{\underline{1}} + \Big\{^n_2 \Big\}x^{\underline{2}} + \cdots + \Big\{^n_n \Big\}x^{\underline{n}}$

It looks so similar to $n2^{n-1} = 1\binom{n}{1}+2\binom{n}{2}+ \cdots + n\binom{n}{n}$ but I have no idea how to prove it

Any ideas?

What is $\left\{\begin{array}{c}n\\k\end{array}\right\}$ ??

Tonio