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Question: The total number of squares of any size (side being natural numbers) in a rectangle of m x n (m \< n...

The total number of squares of any size (side being natural numbers) in a rectangle of m x n (m < n) (m, n ∈ N) is

A

m6\frac { \mathrm { m } } { 6 } (n + 1) (3n - m + 1)

B

m2\frac{m}{2} (m + 1)

C

(m+1)(m+2)4\frac{(m + 1)(m + 2)}{4}

D

None of these

Explanation

Solution

Number of squares = m x n + (m - 1) (n - 1) + (m - 2) (n - 2) +..... + 1. (n - m + 1)

= ∑k=1m(m+1−k)(n+1−k)\sum_{k = 1}^{m}{(m + 1 - k)(n + 1 - k)}

= m6(m+1)(3n−m+1)\frac{m}{6}(m + 1)(3n - m + 1)