Question

Find the number of ways in which two small squares can be selected on the normal chessboard if they are not in the same row or same column.

Answer 1

Hint: We will draw a normal chessboard which has 8 rows and 8 columns and this means that there are 64 small squares in a chessboard. So for selecting the first square we will have 64 squares to select from and then for selecting the second square we will have the number of small squares left to select from.

Complete step-by-step answer:
We will draw a normal chessboard having 8 rows and 8 columns.

			*
			*
			*
*	*	*	1	*	*	*	*
			*
			*		2
			*
			*

So in the figure, 1 represents the 1st square and hence we can select the 1st square in 64 ways.

Now it is given in the question that the two small squares are not in the same row or same column. Hence the stars represent the same row and column in which the second square cannot be there.
2 in the figure represents the 2nd square. So the number of squares left $$=64-8-7=49$$. Hence we can select the 2nd square in 49 ways.
So from product rule we get the number of ways to select two small squares $$=64\times 49........(1)$$.
But we may have selected the 1st square in the table where we had written 2 and the 2nd square where we had written 1. Hence we have counted twice the same squares. So to get the final number of ways we divide equation (1) by 2 and hence we get,
Number of ways $$={\displaystyle \frac{64\times 49}{2}}=1568$$.

Note: Knowing the number of rows and columns of the normal chessboard is the key here. Also drawing the figure is very important because things get clearer. We in a hurry can take the number of squares left to be equal to 48 but here we need to understand that one square we have already counted and hence we get 49.