<p>In the expansion of <span>\((1 + 3x + 2x^2)^6\)</span>, find the coefficient of <em>x</em><sup>11</sup>.</p>
Step-by-Step Solution
Key Concept: Rewrite (1 + 3x + 2x²)⁶ as [(1 + 3x + 2x²)]⁶ and use the multinomial theorem: when expanding, you need terms where powers of x sum to 11. This requires systematically finding all combinations of exponents from individual factors that yield x¹¹.
<p><strong>Step 1:</strong> Use multinomial expansion. In (1 + 3x + 2x²)⁶, we select from each of the 6 factors either 1, 3x, or 2x². If we choose 1 exactly a times, 3x exactly b times, and 2x² exactly c times, then a + b + c = 6.</p><p><strong>Step 2:</strong> The power of x obtained is 0·a + 1·b + 2·c = b + 2c. We need b + 2c = 11.</p><p><strong>Step 3:</strong> From a + b + c = 6 and b + 2c = 11, we get a = 6 - b - c and b = 11 - 2c. For valid solutions: b ≥ 0 gives 11 - 2c ≥ 0, so c ≤ 5.5, thus c ≤ 5. Also a ≥ 0 gives 6 - (11 - 2c) - c ≥ 0, which gives c ≥ 5.</p><p><strong>Step 4:</strong> Therefore c = 5, which gives b = 11 - 10 = 1 and a = 6 - 1 - 5 = 0.</p><p><strong>Step 5:</strong> The multinomial coefficient is 6!/(0!·1!·5!) = 720/(1·1·120) = 6. The term contribution is: 6 · 1⁰ · (3x)¹ · (2x²)⁵ = 6 · 3 · 2⁵ · x¹¹ = 6 · 3 · 32 · x¹¹ = 576x¹¹.</p><p><strong>∴ Coefficient of x¹¹ = 576</strong></p>
Correct Answer: 576