Sum of areas of rectangles hiding in a grid of mxn squares

by luimichael, Oct 24, 2007, 1:27 PM

Denote $ S(m,n)$ as the sum of areas of all possible rectangles hiding in a grid of m-by-n squares.
Then it can be shown that
1. $ S(1,n) = \frac {n(n + 1)(n + 2)} 6 = C(n + 2,3)$;
2. $ S(m,1) = \frac {m(m + 1)(m + 2)} 6 = C(m + 2,3)$;
3. $ S(m,n) = S(m - 1,n) + S(1,n)*C(m + 1,2)$; and
4. $ S(m,n) = C(m + 2,3)*C(n + 2,3)$. :)
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Still have no idea on how to explain the occurrence of the binomial coefficients C(m+2,3) and
C(n+2,3) in the answer.

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