Ic = 2*m*r^2
Ip = 4*m*r^2
Io =Ic+Ip = 6*m*r^2
L = ωo*Io = 6*m*r^2*ωo
angular momentum L is conserved
ω1 = L/Ip = 6*m*r^2*ωo/(2*m*r^2) = 3*ωo
I think I misunderstood the question at the beginning and assumed that the carousel is massless, but if it has mass = 4m without the 4 people this...
4 persons each with mass m stand out on the edge of the carousel that rotates with angular velocity W0. carousel has mass 4m, radius r and inertia I = 2mr^2. The 4 persons then go all the way to the center of the carousel.
Show that the final angular velocity W1 = 3W0