Question

Two smallest squares are chosen one by one on a chessboard. The probability that they have a side in common is:

• A. \(\frac{1}{9}\)
• B. \(\frac{2}{7}\)
• C. \(\frac{1}{18}\)
• D. \(\frac{5}{18}\)

Hint:

To calculate probabilities in combinatorics, count the total number of possible outcomes and favorable outcomes, then divide the two.

Answer 1

The Correct Option is C

Step 1: The problem asks for the probability that two randomly chosen smallest squares on a chessboard share a side. A standard chessboard consists of \(8 \times 8\) squares, which means there are 64 total small squares.

Step 2: The number of possible pairs of squares is given by \(\binom{64}{2}\).

Step 3: The number of favorable outcomes where two squares have a side in common is based on the number of adjacent squares. Each square has up to 4 adjacent squares (except for the edge squares), and the total number of such adjacent square pairs is fewer than 64, as we are restricted to squares that share a side.

Step 4: By counting the adjacent pairs (both horizontally and vertically) and dividing by the total number of pairs, we obtain the probability \(\frac{1}{18}\).