1. Given data

• Masses of the two stars:
  $m_1 = 3M_S, \qquad m_2 = 6M_S$
• Distance between them:
  $a = 9R$
• We need the orbital period of their circular motion about the common center of mass, say $P = nT$.
• $T$ is the Earth's orbital period around the Sun at radius $R$.

2. Use the two-body orbital period formula

For two masses separated by distance $a$, revolving in circular orbits under mutual gravitation, the period is

$P = 2\pi \sqrt{\frac{a^3}{G(m_1+m_2)}}$

Here,

$m_1+m_2 = 3M_S+6M_S = 9M_S$

and

$a=9R$

So,

$P = 2\pi \sqrt{\frac{(9R)^3}{G(9M_S)}}$

$P = 2\pi \sqrt{\frac{729R^3}{9GM_S}}$

$P = 2\pi \sqrt{\frac{81R^3}{GM_S}}$

$P = 9\cdot 2\pi \sqrt{\frac{R^3}{GM_S}}$

3. Relate with Earth's orbital period

For Earth revolving around the Sun,

$T = 2\pi \sqrt{\frac{R^3}{GM_S}}$

Therefore,

$P = 9T$

Hence,

$n=9$

4. Comparison with stored answer

Stored correct answer: $9$
Derived answer: $9$

They match.