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I have N people trying to find a common day for appointment in a period of D days. Each person chooses F favorite days in which he's available during that period.

What is the probability that they will find a matching day for everybody, as function of N,D and F?

A trivial example would be N=2 persons choosing F=1 favorite day over D=7 days. The probability will be P(2,7,1)=1/7.

How to generalize this formula? I'm having trouble combining probabilities I think.

Also, I don't know if this problem can be reframed into a common mathematical problem

P.S. as I'm writing this I'm trying to understand if this is the same problem expressed here

Probability of N people choosing at least one of the same number when simultaneously choosing a set of X distinct numbers from a set of K numbers?

Glasnhost
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  • If everybody chooses his days uniformly at random from the list of available days, and the choices of the various people are independent, then yes, they are the same question. – saulspatz May 08 '21 at 15:15
  • however, the answer makes an assumption that I don't understand, in my case it will be D-N*F>=0, which doesn't make sense – Glasnhost May 08 '21 at 15:20
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    I'm sorry. I misread the other question. It is different. You'll have to use the principle of inclusion and exclusion to answer your question. – saulspatz May 08 '21 at 15:25
  • This is easy enough if there are just two people. The conceptual jump comes at $3$ people. Try doing it when $3$ people each choose $2$ days from $n$. What if they each choose $3$ days? Once you can answer the general question for $3$ people, you'll be able to answer it for any number. – saulspatz May 08 '21 at 15:59

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