Question

The coefficient of \( x^5 \) in the expansion of \( (3 + x + x^2)^6 \) is:

• A. 18
• B. 540
• C. 1620
• D. 2178

Hint:

For multinomial expansions, express terms in the form \( x^b (x^2)^c \) and solve for valid \( (b, c) \) pairs summing to the required exponent.

Answer 1

The Correct Option is D

We are asked to find the coefficient of \(x^5\) in the expansion of \((3 + x + x^2)^6\). Step 1: Apply the Multinomial Theorem The expansion of \((a + b + c)^n\) is given by the multinomial expansion: \[ (a + b + c)^n = \sum_{i+j+k=n} \binom{n}{i,j,k} a^i b^j c^k \] In our case, the expression is \((3 + x + x^2)^6\), so we have \(a = 3\), \(b = x\), and \(c = x^2\). The general term in the expansion will be: \[ \binom{6}{i,j,k} 3^i x^j (x^2)^k \] This simplifies to: \[ \binom{6}{i,j,k} 3^i x^{j+2k} \] Step 2: Find the Values of \(j\) and \(k\) for \(x^5\) We need the exponent of \(x\) to be 5, so: \[ j + 2k = 5 \] We consider the possible values of \(j\) and \(k\) that satisfy this equation: - If \(k = 2\), then \(j = 1\). - If \(k = 1\), then \(j = 3\). - If \(k = 0\), then \(j = 5\). Step 3: Calculate the Corresponding Coefficients For each valid pair \((j, k)\), we substitute into the general term and calculate the coefficient: \begin{itemize} \item For \(k = 2\), \(j = 1\), the term is: \[ \binom{6}{3,1,2} 3^3 x^5 = 60 \times 27 x^5 = 1620 x^5 \] \item For \(k = 1\), \(j = 3\), the term is: \[ \binom{6}{2,3,1} 3^2 x^5 = 60 \times 9 x^5 = 540 x^5 \] \item For \(k = 0\), \(j = 5\), the term is: \[ \binom{6}{1,5,0} 3^1 x^5 = 6 \times 3 x^5 = 18 x^5 \] \end{itemize} Step 4: Total Coefficient of \(x^5\) The total coefficient of \(x^5\) is the sum of the coefficients from each valid term: \[ 1620 + 540 + 18 = 2178 \] Thus, the coefficient of \(x^5\) is \(2178\).