A girl of mass M stands on the rim of a friction less merry-go-round, of radius R and rotational inertia I ,…

The initial angular momentum of the system is zero. The final angular momentum of the girls-plus-merry-go-round is $\left(I+M{R}^2\right)\omega$

The final angular momentum we associate with the thrown
 rock is negative  $\Rightarrow -mRv$

$\Rightarrow mRv=\left(I+M{R}^2\right)\omega$

$\Rightarrow \frac{mRv}{\left(I+M{R}^2\right)}=\omega$

$\Rightarrow R\omega =\frac{mv{R}^2}{\left(I+M{R}^2\right)}$