An Alternate Approach to Alternating Sums: A Method to DIE for.
would you be inclined to try to prove these by counting arguments? Probably not. We confess that there was a time that when we saw alternating sum identities like the ones above, we would attack it with noncombinatorial proof techniques, like induction or generating functions. After all, how can an object be added a negative number of times? Perhaps it can be tackled using the Principle of InclusionExclusion (abbreviated P.I.E.), but that brings another level of complexity to the combinatorial proof. But now we have an alternate opinion. Now when we see an alternating sum, we can usually prove it combinatorially with a method that is as easy as pie and even easier than P.I.E.! As we’ll demonstrate, this technique offers new insights to identities involving binomial coefficients, Fibonacci numbers, derangements, and other combinatorial structures.