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May 2nd, 2018, 06:08 AM  #1 
Newbie Joined: May 2018 From: Germany Posts: 1 Thanks: 0  Find the probability for the event 
May 5th, 2018, 04:15 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
The "complement event" is that there exist at least one empty car and the "PoincareSylvester formula" says that the probability of event A or B (or both) happening is the probability that A happens plus the probability that B happens minus the probability they both happen.

May 5th, 2018, 08:59 AM  #3 
Senior Member Joined: Sep 2015 From: USA Posts: 2,122 Thanks: 1102 
I think the hint in this case leads to a method much harder than necessary. I guess what they are trying to make you do is come up with the probability of the union of $k$ cars being empty $k \in 1,2,\dots N1$ A much much easier approach is to note that in order to maintain at least 1 passenger per car there are $K1$ "slots" between the $K$ passengers that are to be filled with $N1$ "walls". This does assume that passengers are identical which in this problem they essentially are as their probabilities of choosing a car are all identical. So the problem is to determine how many assignments of walls to slots exist and this is simply $n = \dbinom{K1}{N1}$ 

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