Calculate the coefficient at $π₯^3π¦^4π§^5$ in decomposition $(π₯ + π¦ β π§)^{12}$ [closed]
Calculate the coefficient at $π₯^3π¦^4π§^5$ in decomposition $(π₯ + π¦ β π§)^{12}$
If I understood correctly, it needs to be decomposed by Binomial theorem. Thank you for writing your steps and answering the problem.
Solution 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 $C$.
$$12!(-1)^5=C\cdot3!\cdot4!\cdot5!$$