2006 UNCO Math Contest II Problems/Problem 11

Problem

Call the figure below a "$4$-tableau" shape. Determine the number of rectangles of all sizes contained within this shape. Note that a square is considered a rectangle, and a $2\times 1$ rectangle is considered different from a $1\times 2$. Express your answer as a binomial coefficient and explain the significance of your expression. Generalize, with proof, to an "$n$-tableau" shape.

[asy] for(int j=0;j<5;++j){ draw((0,j)--(min(j+1,4),j),black); draw((j,max(0,j-1))--(j,4),black);  } filldraw((2,2)--(2,3)--(1,3)--(1,2)--cycle,blue); filldraw((2,2)--(3,2)--(3,3)--(2,3)--cycle,blue); [/asy]


Solution

$\binom{7}{4}$ in general $\binom{n+3}{4}$

See Also

2006 UNCO Math Contest II (ProblemsAnswer KeyResources)
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