Product of r Choose m with m Choose k/Proof 2

Proof
Consider the trinomial coefficient:
 * $\dbinom r {k, m - k, r - m}$

We use Multinomial Coefficient expressed as Product of Binomial Coefficients:
 * $\dbinom {k_1 + k_2 + k_3} {k_1, k_2, k_3} = \dbinom {k_1 + k_2} {k_1} \dbinom {k_1 + k_2 + k_3} {k_1 + k_2}$

and substitute as appropriate for $k_1, k_2, k_3$.

We have:

Similarly:

The result follows on equating the two expressions.