<p>If \({}^{n+1}C_{r+1} : {}^nC_r : {}^{n-1}C_{r-1} = 11:6:3\), then \(nr =\)</p>
Step-by-Step Solution
Key Concept: Use the ratio property of binomial coefficients by expressing consecutive binomial coefficients in terms of n and r, then set up equations from the given ratio to solve simultaneously.
<p><strong>Step 1:</strong> Express the binomial coefficients in simplified form.</p><p>$$\frac{^{n+1}C_{r+1}}{^nC_r} = \frac{(n+1)!/(r+1)!(n-r)!}{n!/r!(n-r)!} = \frac{n+1}{r+1}$$</p><p>$$\frac{^nC_r}{^{n-1}C_{r-1}} = \frac{n!/r!(n-r)!}{(n-1)!/(r-1)!(n-r)!} = \frac{n}{r}$$</p><p><strong>Step 2:</strong> Apply the given ratio conditions.</p><p>From $^{n+1}C_{r+1} : ^nC_r = 11:6$:</p><p>$$\frac{n+1}{r+1} = \frac{11}{6}$$</p><p>$$6(n+1) = 11(r+1)$$</p><p>$$6n + 6 = 11r + 11$$</p><p>$$6n - 11r = 5 \quad \text{...(1)}$$</p><p><strong>Step 3:</strong> From $^nC_r : ^{n-1}C_{r-1} = 6:3$:</p><p>$$\frac{n}{r} = \frac{6}{3} = 2$$</p><p>$$n = 2r \quad \text{...(2)}$$</p><p><strong>Step 4:</strong> Substitute equation (2) into equation (1).</p><p>$$6(2r) - 11r = 5$$</p><p>$$12r - 11r = 5$$</p><p>$$r = 5$$</p><p><strong>Step 5:</strong> Find n and compute nr.</p><p>$$n = 2r = 2(5) = 10$$</p><p>$$nr = 10 \times 5 = 50$$</p><p>∴ Answer: A</p>
Correct Answer: A