<p>If \(\displaystyle\sum_{r=0}^{n} \frac{r}{{}^nC_r} = \displaystyle\sum_{r=0}^{n} \frac{n^2 - 3n + 3}{2 \cdot {}^nC_r}\), then</p>
Step-by-Step Solution
Key Concept: Recognize that ∑(r/C(n,r)) can be rewritten using the identity r·C(n,r) = n·C(n-1,r-1), and both sides of the equation are sums involving reciprocals of binomial coefficients that require finding specific values or relationships between terms.
<p><strong>Step 1:</strong> Use the identity r·C(n,r) = n·C(n-1,r-1) to rewrite the left side.</p><p>∑(r/C(n,r)) = ∑(r/C(n,r)) involves analyzing individual terms.</p><p><strong>Step 2:</strong> For the equation ∑(r/C(n,r)) = ∑((n²-3n+3)/(2·C(n,r))) to hold, compare structures. The right side has a constant multiplier inside the sum.</p><p><strong>Step 3:</strong> This equality implies that for each term: r/C(n,r) = (n²-3n+3)/(2·C(n,r)), which gives r = (n²-3n+3)/2.</p><p><strong>Step 4:</strong> Since this must hold for the principal terms or weighted average, set r = n/2 (the central term for symmetric consideration): n/2 = (n²-3n+3)/2.</p><p><strong>Step 5:</strong> Multiply by 2: n = n²-3n+3 → n² - 4n + 3 = 0 → (n-1)(n-3) = 0.</p><p>Therefore n = 1 or n = 3. Given the context requires n ≥ 2 for meaningful comparison, n = 3.</p><p>∴ Answer: C (n = 3)</p>
Correct Answer: C