probability with independant variables

hello i need help:

A doorman is trying to get three taxicabs for three different couples. The arrivals of empty cabs has an exponential distribution with mean 2 minutes. assuming independance, what is the probability that he will get al three couples taken care of within six minutes.

in the book the answer is:0.5678

what i did is sum the three independant variables and i got a gamma distribution with mean 2 and alfa=3 then i calculated the integral from 0 to 6 but the answer is wrong

thanks for the help