How Many Rectangles?

Each rectangle is bounded by two pairs of parallel lines, 2 vertical and 2 horizontal. Hence, we may count the two pairs of parallel lines instead of the rectangles (as clearly each pair of pairs determines one unique rectangle). There are 9 vertical lines, hence there are $\binom 9 2$ ways to choose a pair. Same goes for pairs of horizontal lines. Hence the answer is $\binom 9 2^2 = 36^2 = 1296$

For an N x N board...number of rectangles is $\sum _{ r=1 }^{ N }{ { r }^{ 3 } }$

No. of rectangles of any size in a square of $** n \times n = [\frac{n(n+1)}{2}]^2**$

$9C2*9C2=1296$

sum of square n × n grid = 1² + 2² + ... + n² = n(n – 1)(2n – 1)/6

sum of rectangles n × n grid = 1³ + 2³ + ... + n³ = [n(n – 1)/2]²

[8 × 9/2}² = 36² = 1296

Number of rectangles in a figure of 'n*n' is given by n+1C2 * n+1C2 = 1296

9C2 * 9C2 Since there are 9 vertical lines and 2 lines form a rectangle .

On every side of the chessboard we can construct:
Ns(8) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8,
subsides of respectively. 8,7,6,5,4,3,1 elementay subsides associates, so, because: Ns(n) = Sum{(k = 1;n)k} = C(n + 1, 2) = (n +1)n/2, the number of rectangles is:
Nr8x8 = (Ns(8))^2 = [(8 + 1)8/2]^2 = 36^2 = 1296

Ok Guys..it was simple.But i want to modify it a little...how many squares are there on the chess board?/?/? Ans in cmnts below...