Mock AIME 1 2010 Problems/Problem 8

Problem

In the context of this problem, a $\emph{square}$ is a $1 \times 1$ block, a $\emph{domino}$ is a $1 \times 2$ block, and a $\emph{triomino}$ is a $1 \times 3$ block. If $N$ is the number of ways George can place one square, two identical dominoes, and three identical triominoes on a $1 \times 20$ chessboard such that no two overlap, find the remainder when $N$ is divided by 1000.

Solution

Once the pieces are placed, there will be $20-3(3)-2(2)-1=6$ blank spaces. So, we are simply ordering $3$ triominoes, $2$ dominoes, $1$ square, and $6$ blank spaces. That's just $\frac{12!}{3!2!6!} = 55440$ , with last three digits $\boxed{440}$ .

Mock AIME 1 2010 (Problems, Source)
Preceded by Problem 7	Followed by Problem 9
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Mock AIME 1 2010 Problems/Problem 8

Problem

Solution

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