Question

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

• A. $228$
• B. $112$
• C. $108$
• D. $110$

Answer 1

The correct option is 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 verically adjacent squares.
Hence, there are total $56+56=112$ pairs of adjacent squares.