M/M/m Queue birth and death parameters

Question

I would like to determine $\theta_k$ for a $M/M/2$ system. I understand that $\theta_0 = 1$. I also believe that $\lambda_1 = \lambda_2 = \lambda$, and $\mu_1 = \mu$, and $\mu_2 = 2\mu$. Therefore, I think $$\theta_1 = \frac{\lambda}{\mu},\theta_2 = \frac{\lambda^2}{2\mu^2}.\tag{1}$$ However, some solutions I have claim that $$\theta_k = 2\left(\frac{\lambda}{\mu}\right)^k\tag{2}$$ Moreover, from what I understand, the text provides the general case for $M/M/m$, $$\theta_k = \frac{1}{k!}\left(\frac{\lambda}{\mu}\right)^k,\tag{3}$$ for $k = 0,1,\dotsc,m$. It appears that my answer agrees with the book, but that does not mean I am applying it correctly. Some help sorting this out would be appreciated.