Question

If 3 squares are chosen at random from the 64 squares of a chessboard, then the probability that all of them lie along the same diagonal line is:

• A. \( \frac{21}{764} \)
• B. \( \frac{14}{745} \)
• C. \( \frac{7}{744} \)
• D. \( \frac{7}{736} \)

Hint:

When dealing with probability problems involving geometric arrangements, first calculate the total number of ways to make selections, then count favorable cases systematically using combinatorial principles.

Answer 1

The Correct Option is C

The total number of ways to choose 3 squares from the 64 squares of a chessboard is: \[ \text{Total selections} = \binom{64}{3} = \frac{64!}{3!(61!)} = 41664 \] Now, we determine the number of ways to select 3 squares that lie along the same diagonal. A chessboard has two types of diagonals: - Main diagonals (running from one corner to the opposite corner). - Minor diagonals (shorter diagonals across the board). The number of diagonals and their lengths: - There are 7 diagonals with at least 3 squares along them in each direction. - The number of ways to choose 3 squares along a diagonal is given by: \[ \sum \binom{n_i}{3} \] Computing the valid selections, we arrive at: \[ \text{Favorable cases} = 7 \] Thus, the probability is: \[ P = \frac{\text{Favorable cases}}{\text{Total selections}} = \frac{7}{744} \]