InterviewSolution
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InterviewSolution
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`m` equi spaced horizontal lines are inersected by `n` equi spaced vertical lines. If the distance between two successive horizontal lines is same as that between two successive vertical lines, then find the number of squares formed by the lines if `(m < n)` |
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Answer» Let set A of m equi spaced horizontal lines be `l_(1),l_(2), l_(3),.., l_(m)` Let set B of n equi spaced vertical lines be `k_(1),k_(2),k_(3),.., k_(n)` Also let the distance between two consecutive lines be 1 unit. For a square of area 1 sq. units, we must select two consecutive horizontal lines and two consecutive vertical lines which can be done in `(m-1)xx(n-1)` ways. For a square of area 4 sq. units, we must select two lines from set A as `(l_(1),l_(3),(l_(2),l_(4)),.., (l_(m-2),l_(m))` and two from set B as `(k_(1),k_(3)),(k_(2),k_(4)),..,(k_(n-2),k_(n))` `therefore` Number of squares `=(m-2)xx(n-2)` Similarly number of squares of area 9 sq. units `=(m-3)xx(n-3)` and so on. Number of squares of area `(m-1)^(2)` sq. units `=(m-(m-1))xx(n-(m-1))` `therefore` Total number of squares `= underset(r=1)overset(m-1) sum (m-r)(n-r)` |
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