<p>If the sum of the coefficients in the expansion of \((a + b)^n\) is 4096, then the greatest coefficient in the expansion is</p>
Step-by-Step Solution
Key Concept: The sum of all binomial coefficients in (a+b)^n equals 2^n (obtained by setting a=b=1). Once you find n from 2^n = 4096, the greatest coefficient is the middle term coefficient C(n, n/2).
<p><strong>Step 1:</strong> Find n from the sum of coefficients.</p><p>Sum of coefficients in (a+b)^n = C(n,0) + C(n,1) + C(n,2) + ... + C(n,n) = 2^n</p><p>Given: 2^n = 4096 = 2^12</p><p>Therefore, n = 12</p><p><strong>Step 2:</strong> Find the greatest coefficient in the expansion of (a+b)^12.</p><p>For any binomial expansion, the greatest binomial coefficient C(n,r) occurs at the middle term.</p><p>When n = 12 (even), the greatest coefficient is C(12, 6)</p><p><strong>Step 3:</strong> Calculate C(12, 6).</p><p>C(12, 6) = 12!/(6! × 6!) = (12 × 11 × 10 × 9 × 8 × 7)/(6 × 5 × 4 × 3 × 2 × 1)</p><p>C(12, 6) = 924</p><p>∴ Answer: A (924)</p>
Correct Answer: A