Question

what is a combinatorial proof?

Answer 1

Here is an example:

If there are 26 people in a committee and if the president and vice president have to be part of the committee, then how many ways can you choose 8 people to be part of this committee?

The answer is that since 2 people are already part of the committee, then there has to be (24 6) ways to form the rest of the committee. Notice something here: Order does not matter. The objects that are being chosen (people) are not distinct (other then the president and vice president). Combinatorial problems like these involve manipulating n objects taken k at a time. Permutations involve order. In this problem, order does not matter for choosing the other 6 people. So what I said above is pretty much a proof itself (maybe not a formal one, but I am pretty sure it is one nonetheless). Hope this helps.