<p>If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is</p>
Step-by-Step Solution
Key Concept: For identical boxes, we need to count partitions of 12 into 3 parts, then find favorable partitions where exactly one part equals 3. The probability equals (number of partitions with exactly one part = 3) / (total partitions of 12 into 3 parts).
<p><strong>Step 1:</strong> List all partitions of 12 into 3 non-negative parts (identical boxes, unordered):</p><p>{0,0,12}, {0,1,11}, {0,2,10}, {0,3,9}, {0,4,8}, {0,5,7}, {0,6,6}, {1,1,10}, {1,2,9}, {1,3,8}, {1,4,7}, {1,5,6}, {2,2,8}, {2,3,7}, {2,4,6}, {2,5,5}, {3,3,6}, {3,4,5}, {4,4,4}</p><p><strong>Step 2:</strong> Count total partitions: 19 partitions</p><p><strong>Step 3:</strong> Identify partitions where exactly one box contains 3 balls (exactly one part = 3):</p><p>{0,3,9}, {1,3,8}, {2,3,7}, {3,4,5}</p><p><strong>Step 4:</strong> Count favorable outcomes: 4 partitions</p><p><strong>Step 5:</strong> Calculate probability = 4/19</p><p>∴ Answer: D (4/19)</p>
Correct Answer: D