PUNE MCA Exam 2015 Question Paper with answers for online practice

Concept:Probability = (favorable sets of 3 squares on same diagonal) / (total ways to choose 3 squares from 64).Explanation:Total ways to choose 3 squares from 64: $\binom{64}{3} = \frac{64 \times 63 \times 62}{6} = 41664$.Count diagonals (both main and anti-diagonals) on an 8×8 board. Diagonal lengths: 1,2,3,4,5,6,7,8,7,6,5,4,3,2,1 (for each direction). Only diagonals with length $L \ge 3$ contribute. For $L=3$ to $7$, there are 4 diagonals each (2 per direction); for $L=8$, there are 2 diagonals (one per direction).Number of ways to choose 3 squares from a diagonal of length $L$ is $\binom{L}{3}$.Compute: $\binom{3}{3}=1$, $\binom{4}{3}=4$, $\binom{5}{3}=10$, $\binom{6}{3}=20$, $\binom{7}{3}=35$, $\binom{8}{3}=56$.Favorable cases = $4 \times (1+4+10+20+35) + 2 \times 56 = 4 \times 70 + 112 = 280 + 112 = 392$.Probability = $\frac{392}{41664} = \frac{7}{744}$ after simplification.Answer: $\frac{7}{744}$