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 $\binom{n+3}{4}$

2006 UNCO Math Contest II (Problems • Answer Key • Resources)
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2006 UNCO Math Contest II Problems/Problem 11

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