White light with wavelengths from 400 nm to 700 nm falls on a grating that has 400 lines in each…

Step 1. The distance. $d = \dfrac{1\ \text{mm}}{400} = 2500\ \text{nm}$.

Step 2. The first order. The violet end is at $\sin\theta = \dfrac{400\ \text{nm}}{2500\ \text{nm}} = 0.160$, which is $9.21^\circ$. The red end is at $\sin\theta = \dfrac{700\ \text{nm}}{2500\ \text{nm}} = 0.280$, which is $16.26^\circ$. The width is $16.26^\circ - 9.21^\circ = 7.05^\circ$.

Step 3. The red end of order 2. $\sin\theta = \dfrac{2 \times 700\ \text{nm}}{2500\ \text{nm}} = 0.560$, so $\theta = 34.1^\circ$.

Step 4. The violet end of order 3. $\sin\theta = \dfrac{3 \times 400\ \text{nm}}{2500\ \text{nm}} = 0.480$, so $\theta = 28.7^\circ$.

Step 5. The overlap. $34.06^\circ - 28.69^\circ = 5.4^\circ$.