The probability of hitting x royals in an n-play machine when drawing to a 4-card royal is combin(n,x) * (1/47)x * (46/47)n-x. For an explanation of the combin(n,x) function visit my section on probabilities in poker. In the case of 3-play the probabilities are as follows:
0 royals: 0.937519
1 royal: 0.061143
2 royals: 0.001329
3 royals: 0.000010