Q49 / 160 · AP EAMCET 2020 (21 Sep Shift 2) · Mathematics

If the number of rectangles formed on a chess board is 1296, then the total number of squares formed on the chess board is

  • A
    202
  • B
    203
  • Correct answer
    C
    204
  • D
    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.

Hence, the correct answer is option: 'C'.

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