Binomial Theorem
Binomial Coefficients
Grade 11
Question:
<p>The value of \(r\) for which \({}^{20}C_r\,{}^{20}C_0 + {}^{20}C_{r-1}\,{}^{20}C_1 + {}^{20}C_{r-2}\,{}^{20}C_2 + \cdots + {}^{20}C_0\,{}^{20}C_r\) is maximum, is __________.</p>
Step-by-Step Solution
Key Concept: Recognize this sum as the coefficient of x^r in the product (1+x)^20·(1+x)^20 = (1+x)^40 using Vandermonde's identity. The maximum coefficient in (1+x)^40 occurs at r = 20 (the middle term).
<p><strong>Step 1:</strong> Recognize the sum structure. The given sum is:</p><p>Σ(k=0 to r) ²⁰C_k · ²⁰C_(r-k)</p><p><strong>Step 2:</strong> Apply Vandermonde's Identity. By Vandermonde's convolution:</p><p>Σ(k=0 to r) ²⁰C_k · ²⁰C_(r-k) = ⁴⁰C_r</p><p>This is the coefficient of x^r in (1+x)^20 · (1+x)^20 = (1+x)^40</p><p><strong>Step 3:</strong> Find maximum coefficient. In the binomial expansion of (1+x)^40, the coefficient ⁴⁰C_r is maximum when r is at the middle position.</p><p>Since the expansion has 41 terms (r = 0 to 40), the maximum occurs at:</p><p>r = 40/2 = 20</p><p><strong>Step 4:</strong> Verify using the property that for (1+x)^n with even n, ⁿC_(n/2) is maximum.</p><p>∴ Answer: <strong>20</strong></p>
Correct Answer: 20