If the number of rectangles formed on a chess | vedclass

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Question

(C) Let the size of the chessboard be $n 	imes n$. The number of rectangles on an $n 	imes n$ grid is given by $\binom{n+1}{2} 	imes \binom{n+1}{2}

Vedclass · Accepted Answer

$204$

Vedclass · Answer

(C) Let the size of the chessboard be $n 	imes n$. The number of rectangles on an $n 	imes n$ grid is given by $\binom{n+1}{2} 	imes \binom{n+1}{2}$.Given $\left(\frac{n(n+1)}{2}ight)^2 = 1296 = (36)^2$.Thus,$\frac{n(n+1)}{2} = 36$,which implies $n(n+1) = 72$,so $n = 8$.The total number of squares on an $n 	imes n$ board is $\sum_{k=1}^{n} k^2$.For $n = 8$,the sum is $1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2$.Using the formula $\frac{n(n+1)(2n+1)}{6}$,we get $\frac{8 	imes 9 	imes 17}{6} = 4 	imes 3 	imes 17 = 204$.Therefore,the correct option is $C$.

If the number of rectangles formed on a chessboard is $1296$,then the total number of squares formed on the chessboard is:

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