For any positive integers m, n with $n\ge m$ and $\left(\underset{m}{n}\right)=n_{C_m}$then $\left(\underset{m}{n}\right)+\left(n\underset{m}{-}1\right)+\left(n\underset{m}{-}2\right)+----+\left(\underset{m}{m}\right)$ is equal to 

=Coeff.of  $x^{m}in{(1+x)}^n+Coeff.of{x}^{m}in{(1+x)}^{n-1}$

 Coeff.of $x^{m}in{(1+x)}^{n-2}+----$

=Coeff.of $x^{m}in\left[{(1+x)}^n+{(1+x)}^{n-1}+---+{(1+x)}^m\right]$

$=coeff.of{x}^{m}in{(1+x)}^m$$\left[1+(1+x)+{(1+x)}^2+---+{(1+x)}^{n-m}\right]$

$=Coeff.of{x}^{m}in{(1+n)}^m\left[\frac{{(1+x)}^{n-m+1}-1}{(1+x)-1}\right]$

$=Co.eff.of{x}^{m}in\left[\frac{{(1+x)}^{n+1}-{(1+x)}^m}{x}\right]$

$\begin{array}{l}=Co.eff.of{x}^{m+1}in\left[{(1+x)}^{n+1}-{(1+x)}^m\right]\\ =n+1_{C_{m+1}}-0=n+1_{C_{m+1}}\end{array}$