Question

If $16$ squares of unit size are selected at random on a chess board, the probability that they form a square of $4\times 4$ is

• A. $\frac{25}{{}^{64}{C}_{16}}$
• B. $\frac{25}{{}^{64}{C}_4}$
• C. $\frac{36}{{}^{64}{C}_{16}}$
• D. $\frac{36}{{}^{64}{C}_4}$

Answer 1

The correct option is A $\frac{25}{{}^{64}{C}_{16}}$

We need to select 4 consecutive squares in horizontal direction and 4 consecutive squares in vertical direction. This can be done in $(8-4+1)\ast (8-4+1)=25$ ways.
Therefore, probability = $\frac{25}{{}^{64}{C}_{16}}$