AIME II 2010 #11

[stupidityismygam]

Define a T-grid to be a 3x3 matrix which satisfies the following two properties:

Exactly five of the entries are 1's and the remaining four entries are 0's.
Among the eight rows, columns, and long diagonals, no more than one of the eight has all three entries equal.

Find the number of distinct T-grids.

[SumitG]

So...this problem was rather annoying from what I remember.
I think a solution goes something like...
Clearly, there are [unparseable or potentially dangerous latex formula] total possible T-grids. Now, we need to find the amount of grids where either both players in this sort of tic-tac-toe game win once, or player one wins twice.

For the number of grids where both players win once, we see that there are simply [unparseable or potentially dangerous latex formula] possibilities, where the 3,2, and 3s respectively arise from the amount of ways player one can pick a row/column, the amount of ways player two can pick a remaining row/column, and the amount of ways that the three remaining numbers can be arranged.

Now for the number of grids where player one wins twice, we know that either player one wins a row and a column, a row/column and a diagonal, or two diagonals. For the first case, it's clear that there are [unparseable or potentially dangerous latex formula] possibilities. For the second case, there are simply [unparseable or potentially dangerous latex formula] possibilities. For the last case, it's clear that there is just one possibility.

So, now that we have the total number of possible grids, and the number of bad grids, we can subtract them and find the number of "good" grids.
[unparseable or potentially dangerous latex formula]