Question

The number of rectangles which are not squares having common side in a chess board is

• A. ${}^8{C}_2{\times }^8{C}_2-8^2$
• B. ${(}^8{C}_2{\times }^8{C}_2)-\Sigma 8^2$
• C. ${}^9{C}_2{\times }^9{C}_2-8^2$
• D. ${(}^9{C}_2{\times }^9{C}_2)-\Sigma 8^2$

Answer 1

The correct option is D ${(}^9{C}_2{\times }^9{C}_2)-\Sigma 8^2$

As we know that every square is rectangle, so we need to find the total number (including square and rectangle ) and then we will subtract it by number of square in chessboard.
Chessboard has $9$ vertical lines and $9$ horizontal lines and if we will select any two vertical lines and any two horizontal lines, then we will get either square or rectangle ${}^9{C}_2{\times }^9{C}_2$
And number of square in chessboard is $1^2+2^2+.......+8^2$
$\therefore$ number of rectangle is ${}^9{C}_2{\times }^9{C}_2-\sum 8^2$