<p>Find the coefficient of <em>x</em><sup>4</sup> in the expansion of <span>\((1 + x + x^2 + x^3)^{11}\)</span>.</p>
Step-by-Step Solution
Key Concept: Recognize that (1 + x + x² + x³)¹¹ can be rewritten as [(1 + x)(1 + x²)]¹¹ or analyzed using generating functions where we need to count ways to select terms whose powers sum to 4.
<p><strong>Step 1:</strong> Rewrite the base using factorization: (1 + x + x² + x³)¹¹ = [(1 + x)(1 + x²)]¹¹ = (1 + x)¹¹(1 + x²)¹¹</p><p><strong>Step 2:</strong> Expand using binomial theorem:<br/>(1 + x)¹¹ = Σ C(11,r)x^r<br/>(1 + x²)¹¹ = Σ C(11,s)x^(2s)</p><p><strong>Step 3:</strong> For coefficient of x⁴, we need r + 2s = 4 where 0 ≤ r ≤ 11 and 0 ≤ s ≤ 11:<br/>• s = 0: r = 4 → C(11,4)·C(11,0) = 330·1 = 330<br/>• s = 1: r = 2 → C(11,2)·C(11,1) = 55·11 = 605<br/>• s = 2: r = 0 → C(11,0)·C(11,2) = 1·55 = 55</p><p><strong>Step 4:</strong> Add all contributions: 330 + 605 + 55 = 990</p><p>∴ Answer: 990</p>
Correct Answer: 990