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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.

Beneath is my proof graded 2/5.

**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.

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