improve The question Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$ is closed as off-topic while several other questions are closed as duplicates of this question.

This is certainly not an ideal situation.

Since the post has been bumped by an edit to one of the answers I went ahead and added to the post that it does not work for composite $p=4$ and that it works for primes $p=3,5,7$. This is something which OP of such question typically might have tried.

duplicates There are probably several copies of the same question around. (And probably some variations for example asking for some specific type of proof, like combinatorial proof.) It would be nice to choose one of them with good answers as the canonical duplicate target, or even merge some of them if possible.

Looking at questions tagged divisibility+binomial-coefficients, the other questions about this seemed to be closed as duplicates of this one. (I do not count the ones which ask for specific type of proof.)

This one seems to have enough context and some reasonable answers: Proof: if $p$ is prime, and $0<k<p$ then $p$ divides $\binom pk$. However, it is probably meant as proof-verification or proof-explanation question, the OP asks about one specific approach. (Although not all answerers took the question that way.)