<p>The coefficients of \(x^p\) and \(x^q\) (\(p, q \in \mathbb{N}\)) in the expansion of \((1 + x)^{p+q}\) are</p>
Step-by-Step Solution
Key Concept: The binomial coefficient C(n,r) gives the coefficient of x^r in (1+x)^n. Here, coefficient of x^p in (1+x)^(p+q) is C(p+q,p) and coefficient of x^q is C(p+q,q), which are equal by the symmetry property C(n,r) = C(n,n-r).
<p><strong>Step 1:</strong> Apply binomial theorem to (1+x)^(p+q) = Σ C(p+q,r)x^r</p><p><strong>Step 2:</strong> Coefficient of x^p is C(p+q,p) and coefficient of x^q is C(p+q,q)</p><p><strong>Step 3:</strong> Use the symmetry property: C(n,r) = C(n,n-r). Here, C(p+q,p) = C(p+q,(p+q)-p) = C(p+q,q)</p><p><strong>Step 4:</strong> Therefore, both coefficients are equal.</p><p>∴ Answer: The coefficients of x^p and x^q are <strong>equal</strong> (both equal to C(p+q,p) = C(p+q,q))</p>
Correct Answer: A