Combinations

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Question

$The 9 horizontal and 9 vertical lines on an 8 \times 8 chess board form r rectangles$ $(excluding squares) and s squares. The ratio \frac{s}{r} in its lowest terms is:$

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Solution

$\begin{array}{l} No. of squares are S = 1^{2} + 2^{2} + 3^{2} + \dots \dots . . + 8^{2} = \frac{8 (9) (17)}{6} = 204 \\ No. of rectangles r = (^{9} C_{2} \cdot^{9} C_{2}) - 204 = 1092 \\ Hence, \frac{s}{r} = \frac{204}{1092} = \frac{17}{91} \end{array}$

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