The transistor may have minimum Vce(sat) of around 0.2V.
So \(Rc_{total}=\frac{5-0.2}{100mA}=48ohm\)
\(Rc_{ext}=48-(16-3)=35ohm\)
That's for \(Ic=100mA\).
33ohm allows \(Ic>100mA\). I would recommend Rc of 47ohm.
No need to bother about Rb. \(Ib=\frac{V_b-V_{be}}{R_b}\). With 180ohm, it's just a maximum of about 13mA of Ib. 220ohm will give you about 11mA max. 330ohm allows about 7.3mA max. You could still pick a higher Rb from this list. This will further reduce Ib as recommended in previous posts.