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.
There could be several reasons for this. One possibility is that there is an error in your proof or calculation. Another possibility is that there are certain assumptions or conditions that need to be met for the identity to hold, and if those are not satisfied, it can result in a loss. It is important to carefully review your proof and double-check your calculations to identify the cause of the loss.
It depends on the context and the specific application of the identity. In some cases, a 60% loss may be acceptable, while in others it may not be. It is important to consider the significance of the identity in your particular field and determine if the loss is within an acceptable range.
One way to improve your proof is to carefully review and check for any errors or mistakes. You can also seek feedback from colleagues or experts in the field to identify any potential flaws or areas for improvement. Additionally, it may be helpful to explore alternative methods or approaches for proving the identity.
Yes, there are a few common mistakes that can lead to a loss in the proof of this identity. These include incorrect calculations, incorrect assumptions or conditions, and not considering all possible cases or scenarios. It is important to carefully review your proof and check for these potential errors.
Again, it depends on the context and application of the identity. If the loss is within an acceptable range and does not significantly impact the validity of the proof, it may still be usable. However, if the loss is too significant, it may be necessary to revise the proof or explore alternative methods. It is important to carefully consider the implications of the loss and determine if the proof is still applicable in your particular situation.