Given $$\left( 2x + xy - z + \frac{1}{xyz} \right) ^{10}$$
it is asked to caltulate the coeficient of $$ x^6 y^5 z$$
I tried to simplify the formula to only have 3 "parcels" but with no success. the sum of the exponential of $x$, $y$ and $z$ should be $10$ which is not true. Am i missing any rule here?
First, you are looking for $a, b, c, d$ such that $x^a+(xy)^b +z^c+(\frac{1}{xyz})^d = x^6 y^5 z$, where some conditions must be met:
$a+b+c+d=10$, $a+b-d=6$, $b-d=5$ and $c-d=1$.
This is a system of $4$ linear equations with $4$ variables which can be solved leading to $(a,b,c,d)=(1,6,2,1)$.
So you are inspecting the term
$$ \frac{10!}{1!6!2!1!}(2x)^1(xy)^6(-z)^2(\frac{1}{xyz})^1 = 2520\cdot2^1(-1)^2x^6y^5z = 5040\cdot x^6y^5z $$