Find the coefficient of \(x^2 y^2 z^3\) in the expansion of \((4x + 5y - 3z)^7\).
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
\(\dbinom{n}{k_1,k_2,\dots,k_m} = \dfrac{n!}{k_1!k_2!\dots k_m!} \)
\(\begin{array}{|lcll|} \hline x^2 y^2 z^3 \\ n=7,\ m=3 \\ \hline k_1 = 2,\ k_2 = 2,\ k_3 = 3 \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline (4x + 5y - 3z)^7 &=& \dots + \dfrac{7!}{2!2!3!} (4x)^2(5y)^2(-3z)^3 + \dots \\ \hline && \dfrac{7!}{2!2!3!} (4x)^2(5y)^2(-3z)^3 \\\\ &=& \dfrac{7!}{2!2!3!}* 16x^225y^2(-27)z^3 \\\\ &=& \dfrac{4*5*6*7}{4}* 16*25*(-27)x^2y^2z^3 \\\\ &=& -210* 10800*x^2y^2z^3 \\\\ &=& \mathbf{-2268000}x^2y^2z^3 \\\\ \hline \end{array}\)
Source: https://en.wikipedia.org/wiki/Multinomial_theorem