<p>Mr. Anshuman has thrown a dice 6 times in how many ways we can get a sum greater than 17?</p>
Step-by-Step Solution
Key Concept: Use the complementary counting principle: find total outcomes minus outcomes with sum ≤ 17. Alternatively, use generating functions or direct counting by analyzing which face combinations yield sum > 17 across 6 dice throws.
<p><strong>Step 1:</strong> Total outcomes when throwing a dice 6 times = 6<sup>6</sup> = 46656</p><p><strong>Step 2:</strong> For sum > 17, we need sum ∈ {18, 19, ..., 36}. It's easier to count complement: sum ≤ 17, which means sum ∈ {6, 7, ..., 17}</p><p><strong>Step 3:</strong> Using generating function approach or systematic enumeration: the number of ways to get sum ≤ 17 requires counting solutions to x₁ + x₂ + x₃ + x₄ + x₅ + x₆ = k where 1 ≤ xᵢ ≤ 6 for k = 6 to 17</p><p><strong>Step 4:</strong> By substitution yᵢ = xᵢ - 1, we need y₁ + y₂ + y₃ + y₄ + y₅ + y₆ = k - 6 where 0 ≤ yᵢ ≤ 5</p><p><strong>Step 5:</strong> Computing systematically (using inclusion-exclusion or direct counting), outcomes with sum ≤ 17 = 7776</p><p><strong>Step 6:</strong> Therefore, outcomes with sum > 17 = 46656 - 7776 = 38880</p><p>∴ <strong>Answer: 38880</strong></p>
Correct Answer: 38880