Question

The number of ways of selecting two squares on chess board such that they have a side in common is

• A. 224
• B. 112
• C. 56
• D. 68

Answer 1

Answer: (B)

112

There are $64$ squares on the chess board.

Two squares can be selected out of $64$ in ${}^{64}{C}_2$ ways.

Now to select squares such that they have a common side is as follows

in each row, there are $7$ possible pairs of adjacent squares.

Therefore there are $7\times 8=56$ pairs of horizontally adjacent squares.

Similarly, in each column, there are $7$ possible pairs of adjacent squares.

There are $7\times 8=56$ pairs of vertically adjacent squares.

Hence, there are total $56+56=112$ pairs of adjacent squares.