Final answer:
A combinatorial proof of Vandermonde's identity involves considering the ways to choose r objects from a set of m+n objects, either directly or by splitting the set into two parts and counting all possible combinations of choosing objects from each part.
Step-by-step explanation:
Vandermonde's identity is a famous equation in combinatorics that states:
C(m + n, r) = ∑ (C(m, k) * C(n, r - k))
where the sum is over all non-negative integer values of k such that 0 ≤ k ≤ r. To give a combinatorial proof, consider a set with m+n objects, and you want to choose r objects from this set. This can be done directly in C(m + n, r) ways.
Alternatively, split the set into two parts with m and n objects. You could decide to pick k objects from the first part (which can be done in C(m, k) ways) and r-k objects from the second part (which can be done in C(n, r - k) ways). As you can vary k from 0 to r, you add up all these possibilities to get the right-hand side of the Vandermonde's identity.
This proof uses the concept of series expansions from the binomial theorem, which expresses the expansion of the power of a binomial.