Originally Posted by

**surjective** __using induction, I would like to show that__:

$\displaystyle c_{k}=\frac{\ell!(-1)^{k}}{(k!)^{2}(\ell-k)!} \hspace{1cm}(*)$

I have that $\displaystyle c_{0}=1$ for $\displaystyle k=0$. Thus the basis for the induction is established. Now I assume that $\displaystyle (*)$ is true for a fixed $\displaystyle k$ and want to derive the case for $\displaystyle k+1$. I have done the following:

$\displaystyle c_{k+1}=\frac{\ell!(-1)^{k+1}}{((k+1)!)^{2}(\ell-(k+1))!} =\frac{\ell!(-1)^{k}(-1)^{1}}{(k+1)^{2}(k!)^{2}((\ell-k)-1)!}$

I am stuck here. Suggestions would be greatly appreciated.