# Notation in a question on probabilities and particle counting

I'm working through Stephen Barnett's book on quantum information and have come across the following question (1.5, for anyone keeping track at home)

A particle counter records counts with an efficiency $$\eta$$. This means that each particle is detected with probability $$\eta$$ and missed with probability $$1-\eta$$. Let $$N$$ be the number of particles present and $$n$$ be the number detected. Show that

$$P(n|N) = \frac{N!}{(N-n)!n!}\eta^n(1-\eta)^{N-n}$$

a) Calculate P(N|n) for

$$P(N) = e^{-\bar{N}}\frac{\bar{N}^N}{N!}$$

b) Calculate P(N|n) for all P(N) equally probable

c) Calculate P(N|n) given only that the mean number of particles present is $$\bar{N}$$.

I've so far been able to get the first identity, plus part $$(a)$$.

My question is how I'm supposed to interpret $$P(N)$$, as written in question $$(b)$$. The use of capital $$N$$ as well as its use in part $$(a)$$ suggests that it could be "probability that $$N$$ particles are present". However if this the case then it can't be possible for $$P(N)$$ to be constant, since $$N$$ can be any natural number (infinitely many), and the sum of probabilities must be $$1$$. Maybe I'm just supposed to assume that the possible number of particles has an upper bound?

What am I missing here?