If in ∆ABC, ∠C = 105°,

We know, $A+B+C=\pi$

$\therefore A=\pi-(\mathrm{B}+\mathrm{C})$

$\Rightarrow A=180^{\circ}-\left(45^{\circ}+105^{\circ}\right)=30^{\circ}$

Now,

According to sine rule, $\frac{a}{\sin \mathrm{A}}=\frac{b}{\sin \mathrm{B}}$.

$\Rightarrow \frac{2}{\sin 30^{\circ}}=\frac{b}{\sin 45^{\circ}} \quad\left(\because a=2, \angle B=45^{\circ}\right)$

$\Rightarrow \frac{2}{\frac{1}{2}}=\frac{b}{\frac{1}{\sqrt{2}}}$

$\Rightarrow 4 \times \frac{1}{\sqrt{2}}=b$

$\Rightarrow b=2 \sqrt{2}$