The multinomial formula as three Pochhammer rising factorials

Question

I need to describe:

$${n \choose k,0,l,0,m}$$

as three rising factorials. How can I do this?

As far as I know I can delete zero's, so it would be:

$${n \choose k,l,m}=\frac{n!}{k!l!m!},$$

where $k+l+m=n$.

Rising factorial can be written as:

$$\frac{(n+k-1)!}{(n-1)!}$$

But what's next?

Answer 1

Provided $k+l+m=n$, the multinomial coefficient satisfies \begin{align*} \binom{n}{k,l,m} &= \binom{n}{k}\binom{n-k}{l}\binom{n-k-l}{m} \\ &= \frac{1}{k!} (n-k+1)^{(k)} \times \frac{1}{l!} (n-k-l+1)^{(l)} \times \frac{1}{m!} (n-k-l-m+1)^{(m)}. \end{align*}

Could this be what you're after?