Multinomial Term

A trinomial raised to a power expands by the multinomial theorem, where each term arises from distributing the exponent across the three parts, an extension of the binomial idea tested in JEE Advanced. The general term of (1 + x + x^2)^3 is (3! / (a! b! c!)) 1^a x^b (x^2)^c = (3!/(a! b! c!)) x^{b + 2c} with a + b + c = 3. We need the power b + 2c = 4 together with a + b + c = 3. Enumerate non-negative solutions: (b, c) = (0, 2) gives a = 1 and multinomial coefficient 3!/(1! 0! 2!) = 3; (b, c) = (2, 1) gives a = 0 and coefficient 3!/(0! 2! 1!) = 3. Summing, 3 + 3 = 6. Hence the coefficient of x^4 is 6. Option 3 captures only one of the two contributing combinations. Option 9 double counts an invalid arrangement. Option 1 ignores the multinomial coefficients altogether. The two valid (a, b, c) triples must both be included. Plausibility check: by symmetry of (1 + x + x^2)^3 about x^3, the coefficient of x^4 equals that of x^2, and a direct expansion confirms both are 6, validating the count.