<p>Find the coefficient of \(a^6 b^6\) in the expansion of \(\left(a^2 - \dfrac{b}{a}\right)^{12}\).</p>
Step-by-Step Solution
Key Concept: In the binomial expansion of (a² - b/a)¹², the general term is C(12,r)(a²)^(12-r)(-b/a)^r = C(12,r)(-1)^r a^(24-3r) b^r. To get a⁶b⁶, match the exponents: 24-3r=6 and r=6.
<p><strong>Step 1:</strong> Write the general term in the expansion of (a² - b/a)¹²</p><p>T_{r+1} = C(12,r)(a²)^(12-r)(-b/a)^r</p><p><strong>Step 2:</strong> Simplify the general term</p><p>T_{r+1} = C(12,r)(-1)^r a^(24-2r) · b^r · a^(-r)</p><p>T_{r+1} = C(12,r)(-1)^r a^(24-3r) b^r</p><p><strong>Step 3:</strong> Match exponents with a⁶b⁶</p><p>For b^r: r = 6</p><p>For a^(24-3r): 24 - 3(6) = 24 - 18 = 6 ✓</p><p><strong>Step 4:</strong> Calculate the coefficient</p><p>Coefficient = C(12,6)(-1)⁶ = C(12,6) × 1</p><p>C(12,6) = 12!/(6!·6!) = (12×11×10×9×8×7)/(6×5×4×3×2×1) = 924</p><p><strong>∴ Answer: 924</strong></p>
Correct Answer: 924