<p>The probability that all the squares in any column are of same color and that of a row are of alternating color is</p>
Step-by-Step Solution
Key Concept: This problem requires understanding that column uniformity (all same color) and row alternation (alternating colors) are simultaneously constraining conditions that determine a unique coloring pattern. You must recognize that these two conditions together force a specific structure: each column must be monochromatic, and adjacent columns must have opposite colors.
<p><strong>Step 1:</strong> Identify the constraints. For an n×n board: (i) All squares in any column must be the same color, (ii) Colors in each row must alternate.</p><p><strong>Step 2:</strong> Analyze compatibility. If column 1 is all color A, then column 2 must be all color B (to alternate in rows), column 3 must be all color A, etc. This alternating pattern is forced.</p><p><strong>Step 3:</strong> Count valid colorings. There are exactly 2 valid configurations: (i) Odd columns = color A, even columns = color B, or (ii) Odd columns = color B, even columns = color A.</p><p><strong>Step 4:</strong> Calculate probability. Total possible colorings of an n×n board with 2 colors = 2^(n²). Valid colorings satisfying both conditions = 2.</p><p><strong>Step 5:</strong> Therefore, probability = 2/2^(n²) = 2^(1-n²) = 1/2^(n²-1).</p><p>For a standard 2×2 board: probability = 2/2⁴ = 2/16 = 1/8.</p><p>∴ Answer: D</p>
Correct Answer: D