When the NPN emitter is grounded Vbe is ~0.7V and base resistor is easy to calculate:
(5V - 0.7V)/20ma = Rbase
I would think that if you were going to drive the base with 20mA, you might as well leave out the 2N2222. 
When the NPN emitter is grounded Vbe is ~0.7V and base resistor is easy to calculate:
(5V - 0.7V)/20ma = Rbase
I would think that if you were going to drive the base with 20mA, you might as well leave out the 2N2222.