2007 AIME I Problems/Problem 10

Problem

In a 6 x 4 grid (6 rows, 4 columns), 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.

Solution

Casework shows that:

If column 1 and column 2 do not share any two filled squares on the same row, then there are $(6\choose3)$ combinations. The same goes for columns 3 and 4, so we get $(6\choose3)^2 = 400$ .
If column 1 and column 2 share 1 filled square on the same row, then there are $\ldots$ .

Thus, there are $400 + 1080 + 360 + 20 = 1860$ number of shadings, and the solution is $860$ .

2007 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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2007 AIME I Problems/Problem 10

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