Answer
${\bf 3.31\times 10^{12}}\;\rm photon$
Work Step by Step
The dose equivalent in sieverts (Sv) is related to the dose in grays (Gy) and the RBE by:
$$
\text{Dose in Sv} = \text{Dose in Gy} \times \text{RBE}
$$
Solve for the dose in Gy:
$$
\text{Dose in Gy} = \frac{\text{Dose in Sv}}{\text{RBE}}
$$
Substitute the given
$$
\text{Dose in Gy} = \frac{0.30 \times 10^{-3} \, \text{Sv}}{0.85} =\bf 3.53 \times 10^{-4} \, \rm {Gy}
$$
Since 1 Gy is equivalent to 1 Joule per kilogram, the dose in Gy represents the energy absorbed per kilogram of tissue.
$$
\text{Dose in Gy} = \bf 3.53 \times 10^{-4} \, \rm {Gy}= \bf 3.53 \times 10^{-4} \, \rm {J/kg}
$$
Only 25% of the body is exposed, so the effective mass exposed is
$$m_{\rm exposed }= 0.25 \times 60 \, \text{kg} = 15 \, \text{kg} $$
Hence, the total energy absorbed by the exposed portion of the body is:
$$
E = \text{Dose in Gy} \times m_{\rm exposed }
$$
Substitute the values:
$$
E = (3.53 \times 10^{-4} ) \times (15 ) =\bf 5.29 \times 10^{-3} \, \rm {J}
$$
We know that each photon has an energy of $ 10 \, \text{keV} $.
So, the Number of Photons Absorbed is given by
$$ N_{\rm photon} = \frac{\text{Total energy absorbed}}{\text{Energy per photon}}$$
$$N_{\rm photon}=\dfrac{E}{E_{\rm photon}}=\dfrac{5.29 \times 10^{-3}}{10 \times10^3\times 1.6\times 10^{-19}}$$
$$N_{\rm photon} = \color{red}{\bf 3.31\times 10^{12}}\;\rm photon$$
