Wine Tasting Problem (Probability) Mr. Kim claims to be a connoisseur of wine. To test his expertise, he is given 8 cards with the names of 8 types of red wine. He is then presented 8 glasses of these wines and, after tasting them, he has to put one card at each glass. To be acknowledged as a connoisseur of wine, he should guess at least 5 names correctly. Suppose Mr. Kim is not an expert on wines at all. What is the probability that he will be acknowledged as a connoisseur of wine?

I don't think one can use the binomial setting here, as the different choices corresponding to each wine are not pairwise independent (for instance, the last choice is totally determined by the first seven).

It's a problem of counting the number of permutations on a set of cardinality $n$ which have exactly $k$ fixed point (here $n=8$ and $k$ ranges from $5$ to $8$).

We have $P(X=k) = \frac{1}{k!}\sum_{i=0}^{n-k}\frac{(-1)^i}{i!}$

thus $P(X\ge 5)=\frac{1}{360}+\frac{1}{1440}+0+\frac{1}{40320}=\frac{141}{40320}$ if I'm not mistaken.