Suppose you are a teacher who want to assign your students to different clubs. Now a student can join one or more than one club. Now what known to you is there are "n" students and there are "r" clubs.Now how many ways can you do that? If you can solve it using PIE theorem ,then I would be obliged if you also included detailed explanation behind your reasoning and also terribly sorry for my bad English .And I am new to stack so I'm pretty weak at MathJax so feel free to edit. EDIT**:It is possible to get one or more clubs empty**
2026-03-30 08:54:25.1774860865
How to divide students into different groups?
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I'm assuming that every student must belong to a club and that a club can have more than $1$ student(otherwise why would it be called a club). Now, every student has $2$ choices pertaining to a club; join or don't join. Therefore, the number of choices on $r$ clubs is $2^r$. This includes the possibility that the student doesn't join any club. Hence, the number of choices where at least one club must be joined is $2^r-1$. Since each student can choose their clubs independently, the number of assignments of $n$ students to $r$ different clubs is $(2^r-1)^n$.