probability of collision

pandamath

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Nov 17, 2020
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Given n balls, Alice and Bob each randomly chooses 3 different balls. What is the probability that they choose exactly one common ball?
 

Please show us what you have tried, and where you are stuck.

There are several ways you might do this, so I'd want at least to see what method(s) you have learned. I'd probably start by finding the number of ways the choices could be made; and I might look for the probability that there is no collision before answering the question. (Clearly we are to suppose they make their choices independently, without taking any balls away.)
 
Is this an equivalent setup?
There are two boxes containing \(n\) numbered balls(1 to n) . One box for each person.
Alice & Bob each choose three balls from their box.
What is the probability that they have chosen exactly numbered ball in common?
 
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