A bag of mass M hangs by a long thread and a bullet (mass m) comes horizontally with velocity v and gets caught in the bag. Then for the combined system (bag + bullet)

Let V represent the combined system's speed.
Conserving the momentum, 

$$\begin{gathered} \mathrm{mv}=(\mathrm{M}+\mathrm{m})\mathrm{V} \\ \Rightarrow \mathrm{V}=\frac{\mathrm{mv}}{\mathrm{M}+\mathrm{m}} \end{gathered}$$

The kinetic energy of the system = $\frac{1}{2}(\mathrm{M}+\mathrm{m}){\mathrm{V}}^2=\frac{{\mathrm{m}}^2{\mathrm{v}}^2}{2(\mathrm{M}+\mathrm{m})}$

Hence the correct answer is $\frac{{\mathrm{m}}^2{\mathrm{v}}^2}{2(\mathrm{M}+\mathrm{m})}.$