How many rectangles are there in an mxn grid?

Let us consider the following grid with m rows and n columns: 

Number of rectangles with dimension 1X1 = m*n

Number of rectangles with dimension 2X1 = (m-1)*n

Number of rectangles with dimension 3X1 = (m-2)*n

....................

Number of rectangles with dimension mX1 = [(m-(m-1)]*n

Similarly,

Number of rectangles with dimension 1X2 = m*(n-1)

Number of rectangles with dimension 2X2 = (m-1)*(n-1)

Number of rectangles with dimension 3X2 = (m-2)*(n-1)

..............

Number of rectangles with dimension mX2 = [(m-(m-1)]*(n-1)

Increasing the number of columns incrementally by 1, the last set of number of rectangles with n columns are

Number of rectangles with dimension 1Xn = m*1

Number of rectangles with dimension 2Xn = (m-1)*1

Number of rectangles with dimension 3Xn = (m-2)*1

.....................

Number of rectangles with dimension mXn = [(m-(m-1)]*1

Sum (1X1, 2X1, 3X1…..mX1) = n * [m + (m-1) + (m-2) +……1]                = n * m(m+1)/2

Sum (1X2, 2X2, 3X2…..mX2) = (n-1) * [m + (m-1) + (m-2) +……1]         = (n-1) * m(m+1)/2

Sum (1X3, 2X3, 3X3…..mX3) = (n-2) * [m + (m-1) + (m-2) +……1]         = (n-2) * m(m+1)/2

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Sum (1Xn, 2Xn, 3Xn…..mXn) = [(n-(n-1)] * [m + (m-1) + (m-2) +……1]  = 1 * m(m+1)/2

And total number of rectangles will be sum of the sums of rectangles of various sizes

So, S = [m(m+1)/2] * [n + (n-1) + (n-2) +…….1]

S = [m(m+1)/2] * S = [n(n+1)/2]

S = mn(m+1)(n+1)/4

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