Question

If two squares are chosen at random on a chess board, the probability that they have a side in common is

• A. $\frac{1}{9}$
• B. $\frac{2}{7}$
• C. $\frac{1}{18}$
• D. None of these

Answer 1

Answer: (C)

$\frac{1}{18}$

Two square out of $64$ can be selected in ${}^{64}{C}_2=\frac{64\times 63}{2}=32\times 63$ ways
The number of ways of selecting those pairs which have a side in common
$=\frac{1}{2}(4\times 2+24\times 3+36\times 4)=112$
[Since each of the corner squares has two neighbor each of $24$ squares in borders rows, other than corner ones has three neighbor and each of the remaining $36$ squares have four neighbor and in this computation, each pair of square has been considered twice]
Hence required probability $=\frac{112}{32\times 63}=\frac{1}{18}$