A diffraction grating has 300 lines in each millimetre…

Step 1. The distance. $d = \dfrac{1\ \text{mm}}{300} = 3.33\ \mu\text{m} = 3333\ \text{nm}$.

Step 2. The first order. $\sin\theta_1 = \dfrac{\lambda}{d} = \dfrac{520\ \text{nm}}{3333\ \text{nm}} = 0.156$, so $\theta_1 = 8.97^\circ$.

Step 3. The second order. $\sin\theta_2 = 2 \times 0.156 = 0.312$, so $\theta_2 = 18.2^\circ$.

Step 4. The highest order. $\dfrac{d}{\lambda} = \dfrac{3333\ \text{nm}}{520\ \text{nm}} = 6.41$. The highest whole number below this is $6$.