Three smallest squares are chosen randomly on a chess board are the probability that these squares have exactly two corners, but no side common is:
Step-by-Step Solution
Key Concept: Three squares on a chessboard have exactly two corners but no side common when they form a diagonal pattern where two opposite corners of a rectangle are selected along with one corner not adjacent to either. The counting requires identifying valid configurations: 4 corner squares can form ^4C_2=6 pairs of opposite corners, and each pair has 6 valid third squares that share exactly one corner with each endpoint but no sides.
If a shaded square is chosen, 2 additional corner squares must be selected from 4 remaining corner squares, which can be done in $^4C_2 = 6$ ways. The probability is $ rac{^4C_1 \cdot 6^2}{^{64}C_3}$.
Correct Answer: 3