Answer
We can estimate that there are $~~5.07\times 10^{10}~~$ stars in the galaxy.
Work Step by Step
We can express the period in units of seconds:
$T = (2.5\times 10^8)(365)(24)(3600~s)$
$T = 7.884\times 10^{15}~s$
We can use Kepler's Third Law to find the total mass $M$ of all the stars:
$T^2 = (\frac{4\pi^2}{GM})~r^3$
$M = \frac{4\pi^2~r^3}{GT^2}$
$M = \frac{(4\pi^2)~(2.2\times 10^{20}~m)^3}{(6.67\times 10^{-11}~N~m^2/kg^2) (7.884\times 10^{15}~s)^2}$
$M = 1.014\times 10^{41}~kg$
We can find the number of stars:
$\frac{1.014\times 10^{41}~kg}{2.0\times 10^{30}~kg} = 5.07\times 10^{10}$
We can estimate that there are $~~5.07\times 10^{10}~~$ stars in the galaxy.
