A board has $16$ squares (in a $4\times 4$ grid). Out of these $16$ squares, two squares are chosen at random. The probability that they have no side in common is:
Step-by-Step Solution
Key Concept: Total $=\binom{16}{2}=120.$ Adjacent pairs (sharing a side) come in two flavours: horizontal ($3$ per row, $4$ rows) and vertical ($3$ per column, $4$ columns). Subtract from total.
Total ways to pick $2$ squares: $\binom{16}{2}=120.$
Adjacent pairs sharing a side: horizontal $4\cdot 3=12$ and vertical $3\cdot 4=12.$ Total $24.$
Non-adjacent: $120-24=96.$ Probability $=\dfrac{96}{120}=\dfrac{4}{5}.$
Correct Answer: 2