Question:

<p><strong>Step 1:</strong> Write P(X=2) = P(X=3) using the given formula.</p><p>ⁿC₂(1-p)ⁿ⁻²p² = ⁿC₃(1-p)ⁿ⁻³p³</p><p><strong>Step 2:</strong> Simplify by dividing both sides by (1-p)ⁿ⁻³p².</p><p>ⁿC₂(1-p) = ⁿC₃·p</p><p><strong>Step 3:</strong> Expand binomial coefficients.</p><p>[n(n-1)/2](1-p) = [n(n-1)(n-2)/6]·p</p><p><strong>Step 4:</strong> Divide by n(n-1)/2 (assuming n ≥ 3).</p><p>(1-p) = [(n-2)/3]·p</p><p><strong>Step 5:</strong> Solve for the relationship between n and p.</p><p>3(1-p) = (n-2)p</p><p>3 - 3p = np - 2p</p><p>3 = np + p</p><p>3 = p(n+1)</p><p><strong>Step 6:</strong> Find E(X) = np.</p><p>From 3 = p(n+1), we get p = 3/(n+1)</p><p>Therefore, np = n · 3/(n+1) = 3n/(n+1)</p><p>But from 3 = np + p, we directly get: <strong>np = 3 - p</strong></p><p>Since p(n+1) = 3, then p·n = 3 - p, so <strong>E(X) = np = 3 - p</strong></p><p>However, solving completely: n = 5, p = 1/2 satisfies the constraint, giving <strong>E(X) = 5 × 1/2 = 2.5 or 5/2</strong></p><p>∴ Answer: B</p>