Answer
$T = \frac{4\pi r^{1.5}}{\sqrt{G(4M+m)}}$
Work Step by Step
The radius of revolution is $r$
We can find an expression for the force on each star of mass $m$:
$F = \frac{GMm}{r^2}+\frac{Gm^2}{(2r)^2}$
This force provides the centripetal to keep the stars moving in a circle. We can find an expression for the speed $v$:
$F = \frac{GMm}{r^2}+\frac{Gm^2}{(2r)^2} = \frac{mv^2}{r}$
$v^2 = \frac{GM}{r}+\frac{Gm}{4r}$
$v^2 = \frac{4GM}{4r}+\frac{Gm}{4r}$
$v^2 = \frac{G(4M+m)}{4r}$
$v = \sqrt{\frac{G(4M+m)}{4r}}$
We can find an expression for the period:
$T = \frac{distance}{speed}$
$T = \frac{2\pi r}{\sqrt{\frac{G(4M+m)}{4r}}}$
$T = \frac{4\pi r^{1.5}}{\sqrt{G(4M+m)}}$
