If the number of rectangles formed on a chess board is 1296, then the total number of squares formed on the…
If the number of rectangles formed on a chess board is 1296, then the total number of squares formed on the chess board is
202
203
204
205
Solution
It is given that the number of rectangles formed in chess board (Let having size \(n \times n\)) is 1296
\(\begin{aligned}
& \therefore \quad{ }^{n+1} C_2 \times{ }^{n+1} C_2=1296 \\
& \Rightarrow \quad\left(\frac{n(n+1)}{2}\right)^2=1296=(36)^2 \\
& \Rightarrow \quad \frac{n(n+1)}{2}=36 \Rightarrow n(n+1)=72 \Rightarrow n=8 \\
\end{aligned}\)
So, number of squares of any sizes are
\(\begin{aligned}
& n^2+(n-1)^2+(n-2)^2+\ldots \ldots .+1^2 \\
& =1^2+2^2+3^2+\ldots \ldots+8^2 \\
& =\frac{8 \times 9 \times 17}{6}=4 \times 3 \times 17=12 \times 17=204
\end{aligned}\)
Hence, option (c) is correct.