Calculate the coefficient at $^3^4^5$ in decomposition $( + − )^{12}$

Question

Calculate the coefficient at $^3^4^5$ in decomposition $( + − )^{12}$

If I understood correctly, it needs to be decomposed by Binomial theorem. Thank you for writing your steps and answering the problem.

Answer 1

You could also find this using derivatives.

$$(x+y-z)^{12}=\cdots+C x^3y^4z^5+\cdots$$

If you apply $\frac{\partial^{3}}{\partial x^3}\frac{\partial^{4}}{\partial y^4}\frac{\partial^{5}}{\partial z^5}$ to each side, you can solve for $C$.

$$12!(-1)^5=C\cdot3!\cdot4!\cdot5!$$