<p>Given \( {}^{20}C_r \cdot {}^{20}C_0 + {}^{20}C_{r-1} \cdot {}^{20}C_1 + {}^{20}C_{r-2} \cdot {}^{20}C_2 + \cdots + {}^{20}C_0 \cdot {}^{20}C_r \), which equals \( {}^{40}C_r \). Find the value of \( r \) for which this expression is maximum.</p>
Step-by-Step Solution
Key Concept: The sum of products of binomial coefficients equals a single binomial coefficient by Vandermonde's identity: Σ C(20,k)·C(20,r-k) = C(40,r). The maximum of C(40,r) occurs at r = 20 (the middle term of the binomial expansion).
<p><strong>Step 1: Recognize Vandermonde's Identity</strong></p><p>The given sum of products equals C(40,r) by Vandermonde's identity:<br/>Σ C(20,k)·C(20,r-k) = C(40,r)</p><p><strong>Step 2: Find Maximum of C(40,r)</strong></p><p>For any binomial coefficient C(n,r), the maximum value occurs at r = n/2 when n is even.</p><p><strong>Step 3: Apply to C(40,r)</strong></p><p>Since n = 40 (even), the maximum occurs at:<br/>r = 40/2 = 20</p><p><strong>Step 4: Verify using ratio test</strong></p><p>For C(40,r) to be maximum, we need C(40,r) ≥ C(40,r+1) and C(40,r) ≥ C(40,r-1).<br/>The ratio C(40,r+1)/C(40,r) = (40-r)/(r+1).<br/>This equals 1 when 40-r = r+1, giving r = 19.5, so maximum is at r = 20.</p><p>∴ Answer: <strong>20</strong></p>
Correct Answer: 20