+26367 What is the coefficient of \(ab^2c^3\) in \(\left(a + 2b + 3c \right)^6\)?
In general: \(\boxed{\left(a + b + c \right)^n = \sum \limits_{i=0}^{n} \sum \limits_{j=0}^{n-i}\dfrac{n!}{i!j!(n-i-j)!} \cdot a^ib^jc^{n-i-j}}\)
\(\begin{array}{|rlcll|} \hline & n = 6 \\ a^1b^2c^3: & i=1,\ j=2,\ n-i-j = 3 \\\\ & \left(a + 2b + 3c \right)^6 = \dotsb + \dfrac{6!}{1!2!3!} \cdot a^1(2b)^2(3c)^3 + \dotsb \\\\ & \left(a + 2b + 3c \right)^6 = \dotsb + \dfrac{4\cdot 5 \cdot 6}{2} \cdot 2^2 \cdot 3^3 \cdot a^1 b^2 c^3 + \dotsb \\\\ & \left(a + 2b + 3c \right)^6 = \dotsb + 4\cdot 5 \cdot 6 \cdot 2 \cdot 27 \cdot a^1 b^2 c^3 + \dotsb \\\\ & \left(a + 2b + 3c \right)^6 = \dotsb + \mathbf{6480} \cdot a^1 b^2 c^3 + \dotsb \\\\ \hline \end{array} \)
The coefficient of \(\mathbf{ab^2c^3}\) is \(\mathbf{6480}\)
