<p>In how many ways the sum of upper faces of four distinct dice can be six?</p>
Step-by-Step Solution
Key Concept: Find all ordered partitions of 6 into exactly 4 positive integers (each between 1-6), then count distinct arrangements. The constraint that dice are distinct means we count permutations of each valid partition.
<p><strong>Step 1:</strong> Find all partitions of 6 using exactly 4 positive integers, each between 1 and 6 (faces of a die).</p><p>Possible partitions:</p><ul><li><strong>Partition A:</strong> (1,1,1,3) — sum = 6 ✓</li><li><strong>Partition B:</strong> (1,1,2,2) — sum = 6 ✓</li></ul><p><strong>Step 2:</strong> Count permutations for Partition A: (1,1,1,3)</p><p>Permutations = 4!/(3!·1!) = 4 ways</p><p><strong>Step 3:</strong> Count permutations for Partition B: (1,1,2,2)</p><p>Permutations = 4!/(2!·2!) = 6 ways</p><p><strong>Step 4:</strong> Total ways = 4 + 6 = <strong>10</strong></p>
Correct Answer: 10