Question

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

The correct option is C

$\frac{1}{18}$

The explanation for the correct answer.

Solve for the required probability.

Out of $64$ squares, two squares are selected $={}^{64}C_2$ ways $=32x63$ ways

Number of ways of selecting the pairs which have the common side $=\frac{1}{2}$

Favorable cases $=4\left(2\right)+24\left(3\right)+36\left(4\right)=112$

(As each corner has two neighbors, every $24$squares on the border have $3$neighbors, and the remaining $36$ squares have four neighbors)

Hence, the required probability $=\frac{112}{32\times 63}$$=\frac{1}{18}$

Hence, option(C) is the correct answer.