Basic probability counting

Basic probability counting

I'm trying to self-study probability, and I'm trying to solve a problem
(from Walpole) but, since don't have the solution inside don't know how to
validate if my reasoning is fine.
Problem:
There are 16 army troops that need to protect 8 oil towers (numbered from
A to H). They lost contact with each other. Supposing that each army troop
has equal probability to protect any of the tower.
Event A: Which is the probability that at least 4 army troops end
protecting tower A.
Solution:
Given that it doesn't matter the order (is the same to say alpha troop &
beta troop that beta troop & alpha troop) then to count I can use
Combinations. In particular $\binom{16}{8}$ (16 army troops combined
within 8 towers) for the total of sample points.
Now, given that 4 army troops end up protecting tower A it means
$\binom{4}{1}$ (4 army troops within 1 tower) and $\binom{12}{7}$ for the
other towers (there remain 7 towers and 12 troops), so using simple
probability:
$$P(A) = \frac{\binom{4}{1}\binom{12}{7}}{\binom{16}{8}}$$
Is this reasoning right? I'm new to this kind of stuff and I don't know if
I pick the right book but, these counting thing is giving me some
headaches.