+26367 Find the coefficient of \(x^3*y*z^2\) in the expandsion of \((3x - 5y + z)^6\).
Multinomial theorem:
For any positive integer m and any nonnegative integer n, the multinomial formula tells us how a sum with m terms expands when raised to an arbitrary power n:
\(\left( x_1+x_2+\cdots +x_m \right)^n = \sum \limits_{k_1+k_2+\dots+k_m=n} \dbinom{n}{k_1,k_2,\dots,k_m}\cdot x_1^{k_1}\cdot x_2^{k_2}\dots x_m^{k_m}\)
where
\(\dbinom{n}{k_1,k_2,\dots,k_m} = \dfrac{n!}{k_1!k_2!\dots k_m!}\)
Source: https://en.wikipedia.org/wiki/Multinomial_theorem
\(\begin{array}{|lcll|} \hline x^3*y*z^2 \\ n=6,\ m=3 \\ \hline k_1 = 3,\ k_2 = 1,\ k_3 = 2 \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline (3x - 5y + z)^6&=& \dots + \dfrac{6!}{3!1!2!} (3x)^3(-5y)^1z^2 + \dots \\ \hline && \dfrac{6!}{3!1!2!} (3x)^3(-5y)^1z^2 \\\\ &=&\dfrac{6!}{3!1!2!}* 27x^3(-5y)z^2 \\\\ &=& \dfrac{4*5*6}{2}*27*(-5)x^3yz^2 \\\\ &=& -60*135 *x^3yz^2 \\\\ &=& \mathbf{-8100}x^3yz^2 \\ \hline \end{array}\)
