do u actually mean the probability of 20 cards including 5 clubs, 5 diamonds, 5 hearts, and 5 spades?
if so, maybe you could analyze it as follows:
consider one possible sequence of the cards, say we subsequently drew 5 clubs, 5 diamonds, 5 hearts, and 5 spades, since each card will be replaced, so the probability of drawing each card with specific type is 1/13. Thus the probability of getting this sequence is simply (1/13)^20;
given 20 cards, there will be P(20, 20) sequences, now sum them up, we get the P(20, 20)*(1/13)^20.
You could try to evaluate P(20, 20)*(1/13)^20 using R Commander, see if it is close to P = 0.010666 by the experiment.