<s>[INST] Coefficient of $x^4y^3z^3$ in the expansion of $(5x+y-4z)^{10}$ The coefficient of $x^6y^4$ in the expansion of $(2x-3y)^{10}$ is
$$_{10}C_6 \cdot 2^6 \cdot (-3)^4$$
and as for the coefficient of $x^3y^4z^8$ in the expansion of $(x+y+z)^{15}$ is 
$$_{15}C_3 \cdot _{15}C_4 \cdot _{15}C_8 = \frac{15!}{3!4!8!} $$
What would be the the coefficient if the case would be $x^4y^3z^3$ in the expansion of $(5x+y-4z)^{10}$
Thank You
With Respect Umer Selmani
 [/INST] We have that by trinomial expansion
$$(5x+y-4z)^{10} =\ldots+\frac{10!}{4!3!3!} (5x)^4y^3(-4z)^3+\ldots$$
therefore the coefficient for $x^4y^3z^3$ is equal to $-\frac{10!5^34^3}{4!3!3!}$.
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