Let me start with an example from geometry to help with the idea of a combinatorial proof. Suppose we want to find the area of a 3-4-5 triangle.

1. One student calculates the area as $\dfrac{1}{2}\cdot 3\cdot 4$.
2. A second calculates the area as $\sqrt{6(6-3)(6-4)(6-5)}$.

Without any calculations, if we recognize that the first student is using $\dfrac{1}{2}\cdot b\cdot h$ and the second student is using Heron's formula, we can say that both students got the correct answer (and they are the same) WITHOUT actually simplifying both equations.

The idea of a combinatorial proof is that if we answer the SAME question in different ways (correctly of course!) we must get the same answer either way.

For example, in 6.3, this question to be answered in multiple ways is "If you have n people and want to choose a committee of k people, with one person as the president on the committee, how many ways can it be done?" One method leads to an answer of $\displaystyle \binom{n}{k}\cdot k$ and an alternate method gives $\displaystyle n\cdot \binom{n-1}{k-1}$. Since both methods are correct, the two answers must be equal!

As a hint for 6.17 (and 6.30), often subtractions are a hint that negative numbers show up in the binomial theorem. For example, for 6.17, try to write out what the binomial theorem says for expanding $(1 + (-1))^10$ (that is $a = 1$ and $b=-1$ in the Binomial Theorem as written in the textbook).

Hope this helps!