2007 AIME I Problems/Problem 10

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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 $860$.

See also

2007 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
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All AIME Problems and Solutions
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