This is why I don't do math. -AS, commenting on the ways to get to 270 electoral votes
I'm not sure I understand the comment. Is it because there are people who are better or because of the types of problems?
Why is it more interesting to, say, spend hours reading through the history of Winston Churchill, which doesn't yield any 'final result', very much?
VOTING BLOCKS
Anyway, from the massive-size of the result, you can see why "voting blocks" are so important to controlling outcomes. There is just too much uncertainty associated with 51 trillion combinations to bear ... (If fact, I was just about to do a 'voting-block' analysis on the electoral college, when this came up).
JUST UNDERTAUGHT?
By the way, combinatorics is a fascinating field. The problems are far, far more interesting than ... crossword puzzles or sudoku.
FINDING WAYS TO SOLVE (OR DESCRIBE) PROBLEMS
Sadly, I don't remember the polynomial method that Isabel has shown, which appears to be very fundamental.
The insight is that you can use algebra (polynomials) to enumerate combinations. In lingo, such enumerations are done with "generating functions". (I recall 'moment generating functions' for distributions, but not for discreet combinatorics... what do I know?).
It's easy enough to figure-out combinations for things that are all "evenly sized" or the same, like how many combinations of 9 players you can create from a 15 member roster. It's easy enough to figure out the number of ways you can arrange a group of something (2^n or 2n-1), i.e. create "sets" and "subsets" from a given lot. It's easy enough to figure out how to make combinations, taking from one group and from another, dissimilar yet homogeneous group, i.e. assuming your team roster is divided into fullbacks, halfbacks, and forwards...
EXPANDING THE TOOLBOX - WHO COULD REFUSE MORE TOOLS?
However, when you are creating combinations of differently sized things, you need another insight. Would you have been clever enough to come up with the polynomial approach? Sure, in retrospect, it is easy enough to see it as a clever way to add up 'combinations that are similarly sized', because it both (a) enumerates all the combinations and provides an easy way to add up the ones that are 'similar', because their exponents will all be the same. But in prospect, I'm not sure I would have thought of it (without having been taught it, say).
Even so, one would still need a high-level math program, capable of symbolic manipulation, like Mathematica, Maple, or MatLab... (although I think I see a way around that).
Anyway, one also has to appreciate the continuous approximation of the combinations. As best I recall, these can often be a good check on results.
