<p>Coefficient of \(x^{11}\) in the expansion of \((1+x^2)^4(1+x)^7(1+x^4)^{12}\) is</p>
Step-by-Step Solution
Key Concept: Rewrite the product as separate binomial expansions and identify all combinations of terms whose exponents sum to 11: from (1+x²)⁴ pick x^(2a), from (1+x)⁷ pick x^b, from (1+x⁴)¹² pick x^(4c) where 2a + b + 4c = 11.
<p><strong>Step 1:</strong> Identify coefficient of x¹¹ from (1+x²)⁴(1+x)⁷(1+x⁴)¹²</p><p>From (1+x²)⁴: term is C(4,a)x^(2a)</p><p>From (1+x)⁷: term is C(7,b)x^b</p><p>From (1+x⁴)¹²: term is C(12,c)x^(4c)</p><p>Need: 2a + b + 4c = 11 where a ≤ 4, b ≤ 7, c ≤ 12</p><p><strong>Step 2:</strong> Find all valid (a,b,c) combinations</p><p>• c = 0: 2a + b = 11 → (a,b) = (0,11)✗, (1,9)✗, (2,7)✓, (3,5)✓, (4,3)✓</p><p>• c = 1: 2a + b = 7 → (a,b) = (0,7)✓, (1,5)✓, (2,3)✓, (3,1)✓</p><p>• c = 2: 2a + b = 3 → (a,b) = (0,3)✓, (1,1)✓</p><p><strong>Step 3:</strong> Calculate coefficient sum</p><p>C(4,2)C(7,7)C(12,0) + C(4,3)C(7,5)C(12,0) + C(4,4)C(7,3)C(12,0)</p><p>+ C(4,0)C(7,7)C(12,1) + C(4,1)C(7,5)C(12,1) + C(4,2)C(7,3)C(12,1) + C(4,3)C(7,1)C(12,1)</p><p>+ C(4,0)C(7,3)C(12,2) + C(4,1)C(7,1)C(12,2)</p><p>= 6(1)(1) + 4(21)(1) + 1(35)(1) + 1(1)(12) + 4(21)(12) + 6(35)(12) + 4(7)(12)</p><p>= 6 + 84 + 35 + 12 + 1008 + 2520 + 336</p><p>= 4001</p><p>∴ Answer: D</p>
Correct Answer: D