AIME I Problem 10 2007

[zefuri]

In the 6 x 4 grid shown, 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let N be the number of shadings with this property. Find the remainder when N is divided by 1000.

[mathpimp]

Dude that sounds like the Sudoku puzzle from hell, without the numbers.

I'm too tired, and too lazy right now to think about it, maybe later.

[stupidityismygam]

It is mad casework

Consider 4 cases

Case 1: In column 1, and column 2 no two shadings are in the same row

We have [unparseable or potentially dangerous latex formula] choices for the 1st column, then 1 choice for the 2nd column, then [unparseable or potentially dangerous latex formula] choices for the 3rd column, and 1 choice for the 4th column. So the total amount is [unparseable or potentially dangerous latex formula]

Case 2: In column 1 and column 2 1 set of two shadings are in the same row

We have [unparseable or potentially dangerous latex formula] choices for the 1st column, then [unparseable or potentially dangerous latex formula] choices for the 1 shading that is in the same row and [unparseable or potentially dangerous latex formula] for other 2 shadings. In the third column, 1 of the shadings must go in a particular square, and then for the other two shadings there are [unparseable or potentially dangerous latex formula]. Which is a total of [unparseable or potentially dangerous latex formula]

Case 3: In column 1 and column 2 2 sets of shadings are in the same row

We have [unparseable or potentially dangerous latex formula] choices for the 1st column, then [unparseable or potentially dangerous latex formula] choices for the 2 shading that is in the same row and [unparseable or potentially dangerous latex formula] choices for the last shading. In the third column, 2 shading must go in a particular spot, and then there are [unparseable or potentially dangerous latex formula] choices for the last shading. Which is a total of [unparseable or potentially dangerous latex formula]

Case 4: In column 1 and column 2 3 sets of shadings are in the same row

We have [unparseable or potentially dangerous latex formula] choices for the 1st column, and the other columns only have 1 choice. Thus the total is [unparseable or potentially dangerous latex formula].

Summing these totals up gives [unparseable or potentially dangerous latex formula]

So the answer is [unparseable or potentially dangerous latex formula]