# Probability that k people choose the same set of n numbers

How can I calculate the probability that at least 2 people out of kk people choose an identical set of nn numbers? The numbers are positive integers, starting with 11, and every number only appears once. We assume that every person picks their numbers randomly.

Example:
33 people choose 55 numbers between 11 and 2020:

Person 1: 5,8,10,12,135, 8, 10, 12, 13
Person 2: 4,5,6,7,84, 5, 6, 7, 8
Person 3: 5,8,10,12,135, 8, 10, 12, 13

How likely is it that Person 1 and Person 3 picked the same numbers?

I would need the formula in a generic way so that kk and nn can vary. Any help is appreciated!
Bonus: How would I need to modify the formula if I want to know the probability that “at least 3 people” choose the same numbers?

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Are the numbers ordered? Is there a difference between picking 1,2,3,4,51,2,3,4,5 and 5,4,3,2,15,4,3,2,1?
– Kevin Long
2 days ago

no, there would be no difference, the order is irrelevant
– user2037036
yesterday

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Hint: The probability that at least two people pick the same set of numbers is equal to 1−1- the probability that everybody picks different sets of numbers.

Thanks, so how can I calculate the probability that everybody picks different sets of numbers? Unfortunately my math skills are not very sophisticated…
– user2037036
yesterday

@user2037036 Let’s say the first person picks kk numbers. Then the second person can pick kk out of n−kn-k numbers. Do you know how to count this?
– Kevin Long
yesterday