What is the probability that you identify correctly each painter and her painting? > At an art gallery, you enter a room where $6$ paintings are hung on its walls. Four people are in the room. If you know that one of these people is not a painter and the other three are the painters of three of the works exhibited, what is the probability that you identify correctly each painter and her painting? My analysis: First there is only one correct way of identifying each painter with her painting. I know that only $3$ are painters and one is not, so I should identify $3$ distinct persons with $6$ distinct paintings. I assume there should be a permutation of $6P3=120$ and the probability must be $1/120$, but according to the mcq the correct answer is:$1/480$. How should I proceed?

Can be done by matching the painters up with their paintings, one after the other, probability-tree style.

First, you have to pick one of the $3$ painters out of the $4$ people $\frac{3}{4}$. Then, you have to pick the 1 painting they painted out of $6$. Probability of doing this is $\frac{3}{4}\times\frac{1}{6}$. Next, there are $2$ painters left out of $3$ people, and $5$ paintings left. Probability of getting both the first pick correct, and the second is $\frac{3}{4}\times\frac{1}{6}\times\frac{2}{3}\times\frac{1}{5}$.

Therefore, the probability of getting all 3 correct is $\frac{3}{4}\times\frac{1}{6}\times\frac{2}{3}\times\frac{1}{5}\times\frac{1}{2}\times\frac{1}{4} = \frac{1}{480}$