<p>The largest coefficient in the expansion of \((1 + x)^{24}\) is</p>
<p>(a) \(^{24}C_{24}\)</p>
<p>(b) \(^{24}C_{13}\)</p>
<p>(c) \(^{24}C_{12}\)</p>
<p>(d) \(^{24}C_{11}\)</p>
Step-by-Step Solution
Key Concept: In the expansion of (1+x)^n, the largest coefficient occurs at the middle term(s). For even n, the largest coefficient is C(n, n/2). For (1+x)^24, we need C(24, 12).
<p><strong>Step 1:</strong> In the binomial expansion of (1+x)^n, the general term is C(n,r)x^r where r = 0, 1, 2, ..., n.</p><p><strong>Step 2:</strong> The coefficient of each term is C(n,r). These coefficients increase until the middle and then decrease symmetrically.</p><p><strong>Step 3:</strong> For (1+x)^24, we have n = 24 (even). The coefficients are largest at the middle term, which occurs at r = n/2 = 24/2 = 12.</p><p><strong>Step 4:</strong> The largest coefficient is C(24, 12) = 24!/(12!×12!) = 2,704,156.</p><p><strong>Step 5:</strong> This can be verified by noting that C(24,12) > C(24,11) and C(24,12) > C(24,13), since the ratio C(n,r+1)/C(n,r) = (n-r)/(r+1) equals (24-12)/(12+1) = 12/13 < 1 for r = 12.</p><p>∴ Answer: C(24, 12) or 2,704,156</p>
Correct Answer: C