Why did I lose 60% on my proof of Generalized Vandermonde's Identity? What was amiss?

[scherz0]

My tests are submitted and marked anonymously. I got a 2/5 on the following, but the grader wrote no feedback besides that more detail was required. What details could I have added? How could I perfect my proof?

Prove Generalized Vandermonde's Identity, solely using a story proof or double counting. DON'T prove using algebra or induction —  if you do, you earn zero marks.

$\sum\limits_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }.$

Beneath is my proof graded 2/5.

 

I start by clarifying that the summation ranges over all lists of NONnegative integers $(k_1,k_2,\dots,k_p)$ for which $k_1 + \dots + k_p = m$. These $k_i$ integers are NONnegative, because this summation's addend or argument contains $\binom{n_i}{k_i}$.

On the LHS, you choose $k_1$ elements out of a first set of $n_1$ elements; then $k_2$ out of another set  of $n_2$ elements, and so on, through $p$ such sets — until you've chosen a total of $m$ elements from the $p$ sets.

Thus, on the LHS, you are choosing $m$ elements out of $n_1+\dots +n_p$, which is exactly the RHS. Q.E.D.

Edited January 29, 2022 by scherz0