Question

Nine small squares of size $1\times 1$ are chosen at random on a chessboard. What is the probability that they form a big square of size $3\times 3$ ?

• A. $\frac{9}{{}^{64}{C}_9}$
• B. $\frac{36}{{}^{64}{C}_9}$
• C. $\frac{6}{{}^{64}{C}_9}$
• D. None of these

Answer 1

The correct option is B $\frac{36}{{}^{64}{C}_9}$

We can choose 9 squares out of 64 squares in ${}^{64}{C}_9$ ways.
Hence exhaustive number of cases ${=}^{64}{C}_9$
From the figure it is clear that given square of size $3\times 3$

can be formed by using four consecutive horizontal and 4 consecutive vertical lines. which can be done in
${}^6{C}_1{\times }^6{C}_1=36$ ways
Basically you can make 6 squares of size $3\times 3$ in vertical direction and 6 squares of the size $3\times 3$ in horiozntal direction. Hence total $6\times 6=36$ squares can be chosen.
$\therefore$ The required probability $=\frac{36}{{}^{64}{C}_9}$