Question

Prove that \(\sum_{k=0}^{p}\left(\begin{array}{c}m \\\ k\end{array}\right)\left(\begin{array}{c}n \\\ p-k\end{array}\right)=\left(\begin{array}{c}m+n \\ p\end{array}\right)\) for non-negative integers \(m, n\) and \(p\). (This equation is from Exercise 7 in Section 3.10 . There we were asked to prove it by combinatorial proof. Here we are asked to prove it with induction.)

Short answer

The given identity is proven by induction and holds true for all non-negative numbers m, n and p.

Step-by-step solution

Proof for Base Case (p = 0)

For \(p=0\), the identity is \( \binom{m}{0} \binom{n}{0} = \binom{m + n}{0} \). The binomial coefficient \( \binom{a}{0} = 1 \) for all non-negative integers \( a \), so we have \( 1 * 1 = 1 \). Therefore the identity holds true for \( p = 0 \).

Assume the Identity is True for p = k

To use induction, we first assume the formula is true for \( p = k \), i.e., \( \sum_{i=0}^{k} \binom{m}{i} \binom{n}{p-i} = \binom{m + n}{k} \). We take this as the Inductive Hypothesis, but this isn't something we need to prove.

Prove Identity is True for p = k+1

Let's consider \( p = k+1 \). We have \( \sum_{i=0}^{k+1} \binom{m}{i} \binom{n}{p-i} = \sum_{i=0}^{k} \binom{m}{i} \binom{n}{k-i} + \binom{m}{k+1} + \binom{n}{0} = \binom{m+n}{k} + \binom{m+n}{k+1} = \binom{m+n}{k+1} \). This completes the proof by induction.