Coefficient Extraction

Products of binomial powers with the same base are best merged by adding exponents before extracting any coefficient, a streamlined JEE Advanced technique that avoids tedious convolution. Since (1 + x)^4 (1 + x)^6 = (1 + x)^{4+6} = (1 + x)^{10}, the coefficient of x^5 is simply C(10, 5). Evaluating, C(10, 5) = 10! / (5! 5!) = 252. Hence the required coefficient is 252. Option 210 = C(10, 6) corresponds to the wrong power of x. Option 120 = C(10, 3) again targets a different term. Option 45 = C(10, 2) is far from the central term. The crucial simplification is recognising that multiplying the powers adds the exponents, collapsing the product into a single binomial. Alternatively, the Vandermonde identity sum over C(4, k)C(6, 5 - k) for k from 0 to 4 reproduces the same 252, cross-validating the merge and showing why the convolution of the two coefficient sequences agrees with the single merged coefficient. This equivalence is no accident: multiplying generating functions corresponds exactly to convolving their coefficient sequences, which is the algebraic content of the Vandermonde identity. Plausibility check: the coefficient of x^5 should be the central, hence largest, coefficient of (1 + x)^{10}, and 252 is indeed the maximum entry in that row, consistent with the symmetry of Pascal's triangle.