<p>If the coefficients of \(x^{39}\) and \(x^{40}\) are equal in the expansion of \((p + qx)^{49}\), then the possible values of \(p\) and \(q\), respectively, are</p>
Step-by-Step Solution
Key Concept: In the binomial expansion of (p + qx)^49, the general term is C(49,r)·p^(49-r)·(qx)^r. For coefficients of x^39 and x^40 to be equal, set the binomial coefficients and power expressions equal, then use the property that C(n,r) = C(n,n-r) to establish a ratio relationship.
<p><strong>Step 1:</strong> Write the general term in the expansion of (p + qx)^49.</p><p>T_{r+1} = C(49,r)·p^(49-r)·(qx)^r = C(49,r)·p^(49-r)·q^r·x^r</p><p><strong>Step 2:</strong> Identify coefficients of x^39 and x^40.</p><p>Coefficient of x^39: C(49,39)·p^10·q^39</p><p>Coefficient of x^40: C(49,40)·p^9·q^40</p><p><strong>Step 3:</strong> Set them equal and simplify.</p><p>C(49,39)·p^10·q^39 = C(49,40)·p^9·q^40</p><p>C(49,39)·p = C(49,40)·q</p><p><strong>Step 4:</strong> Use C(49,39) = C(49,10) and C(49,40) = C(49,9).</p><p>Also note: C(49,40)/C(49,39) = 10/40 = 1/4</p><p>Therefore: p/q = C(49,40)/C(49,39) = 10/40 = 1/4</p><p>∴ p:q = 1:4 or the possible values are p = 1, q = 4 (and equivalent ratios)</p><p><strong>Answer: ACD</strong></p>
Correct Answer: ACD