The forum discussion centers on the proof of Generalized Vandermonde's Identity, specifically the requirement to use a story proof or double counting method rather than algebra or induction. The original proof submitted received a score of 2/5 due to insufficient detail and clarity regarding the summation and the concept of double counting. Key feedback indicated that the proof needed to explicitly demonstrate how the sum captures all possible selections of m elements from p sets, and the reasoning should have been approached from the left-hand side (LHS) to the right-hand side (RHS) to clarify the argument.
PREREQUISITESMathematics students, educators, and anyone interested in improving their proof-writing skills, particularly in combinatorial mathematics and identity proofs.
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.
$$\sum\limits_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }.$$
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.