Suppose we are given n balls, labeled 1, · · · , n, as well as n boxes, also labeled 1, · · · , n.
We randomly draw one of the n balls and one of the n boxes, and place the ball in the box. This
procedure is repeated n times, until every ball is in a box. Each time the ball is chosen uniformly
from the remaining balls, and the box is chosen uniformly from the remaining boxes.

(a) What is the probability that the ball labeled k ends up in the box labeled k?

(b) What is the expected number of label matches?

(c) What is the variance of the number of label matches?

I think for (a), should it just be 1/n^2? But I think it is not correct, because it is without replacement. Can anyone help me with this? Thanks a lot.