Question

Combinatorics
Use algebraic methods to prove the cancellation identity : (N | K) ( K | M) = (N | M ) (N - M | K - M)

Answer 1

choose numbers, not division

Answer 2

I'm a little late here aren't I :/ Oh well. This is most of the method you need.

Answer 3

\[\begin{aligned}\binom{n}{k}\binom{k}{m}&=\frac{n!}{k!(n-k)!}\cdot\frac{k!}{m!(k-m)!}\\
&=\frac{n!}{m!(n-k)!(k-m)!} \\
&=\frac{n!}{m!(n-m!)}\cdot\frac{(n-m)!}{(k-m)!(n-k)!} \qquad \text{(multiply by } \frac{(n-m)!}{n-m)!})\\
&=\frac{n!}{m!(n-m!)}\cdot\frac{(n-m)!}{(k-m)!(n-m-k+m)!} \qquad \text{add/subtract } m
\end{aligned} \]You should be able to see the finish from here.