A train has got five train cars, each one with N seats. There are 150 passengers who randomly choose one of the cars. What is the probability that everyone will get a seat?

I think that what is asking me is "what is the probability that each wagon is chosen by no more than N passengers"?

The probability of having n<N people in *a* wagon is given by
p_N=\sum\limits_{n=0}^N \binom{150}{n}0.2^n 0.8^{150-n}.

I thought of the answer being p_N^5.. but I think that the events

- wagon a is chosen by no more than N passengers
- wagon b is chosen by no more than N passengers
- wagon c is chosen by no more than N passengers
- wagon d is chosen by no more than N passengers
- wagon e is chosen by no more than N passengers

are far from being independent... So what could I do?
Thanks a lot