Recall the combinatorial interpretation of the binomial coefficients in the binomial theorem: they represent the number of ways to choose k elements from a set of n elements.

Consider how the binomial expansion (a + b)^n can be derived by systematically counting the number of ways to form each term in the expansion using combinations.

Prove the binomial theorem using mathematical induction, carefully analyzing the base case and the inductive step where you assume the theorem holds for (a + b)^n and then prove it for (a + b)^(n+1).