Question

The number of rectangles which are not squares having common side in a chess board is
A. $c_2^8 \times c_2^8 - {8^2}$
B. $(c_2^8 \times c_2^8) - \sum {{8^2}} $
C. $c_2^9 \times c_2^9 - {8^2}$
D. $(c_2^9 \times c_2^9) - \sum {{8^2}} $

Answer 1

Hint: According to the question given in the question we have to determine the number of rectangles which are not squares having a common side in a chess board. So, first of all we have to determine the total number of rectangles in the chess board.
Now, as we know that a square is a rectangle and if count the number of squares in the chess board in the horizontal and vertical line of the chess board.
Now, as we know that if we add two squares then it will form a rectangle hence, we have to determine the number of ways to chose the two squares from the vertical line with the help of the formula which is as mentioned below:

Formula used:
$ \Rightarrow c_r^n = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}...................(A)$
Where, n is the total number of ways or possibility and r is the number of ways required.
Now, same as we have to determine the number of ways to choose the two squares in the horizontal line which can be determined with the help of the formula (A) which is as mentioned above.

Complete step-by-step answer:
Step 1: First of all we have to determine the total number of rectangles in the chess board as mentioned in the solution hint and as we know that there are 9 vertical lines and 9 horizontal lines.
Step 2: Now, as we know that a square is a rectangle so we have to form two squares in the form of a rectangle and then choose them in the total number of squares in the vertical and horizontal lines.
Step 3: Now, we have to determine the number of ways to choose the two squares in the vertical line which can be determined with the help of the formula (A) which is as mentioned in the solution hint. Hence,
 $_{}^2{C_9}$
Step 4: Now same as step 3, we have to determine the number of ways to choose the two squares in the horizontal line which can be determined with the help of the formula (A) which is as mentioned in the solution hint. Hence,
 $_{}^2{C_9}$
Step 5: Now, we have to determine the total number of squares in the chess board which can be obtained by adding all the squares in the chess board as below:
$
   = {1^2} + {2^2} + {3^2} + {4^2} + ............... + {8^2} \\
   = \sum {{8^2}} \\
 $
Step 6: Now, with the help of the solution steps 3, 4 and 5 we can obtain the required number of rectangles by multiplying all of them to each-other and then we have to subtract them by the total number of squares in the chess board. Hence,
$ \Rightarrow c_2^9 \times c_2^9 - \sum {{8^2}} $

Hence, we have obtained the required number of rectangles which are $ c_2^9 \times c_2^9 - \sum {{8^2}} $. Therefore option (D) is correct.

Note:
To determine the required number of rectangles it is necessary that we have to subtract the total number of rectangles in the chess board by the multiplication of the rectangles in the vertical and horizontal plane of the chess board. A square is a rectangle and if count the number of squares in the chess board in the horizontal and vertical line of the chess board.