Need help with this problem. I have an idea on a but not sure if a is right nor what to do with b or c. I went to a poorly planned “gift exchange” All five of us tossed our gifts into a bin, and then picked one at random. It was bad, because there was a good chance someone would receive his or her own gift!
a. What is the probability that nobody received his or her own gift? I was thinking 4!/5!, but not sure.
b. What is the probability that at least one person received his or her own gift?
c. What is the probability that at least two people received their own gifts?
For a, you are looking for the probability of a derangement, which is very close to $\frac 1e$ OEIS gives the exact values. For five, it is $\frac {44}{120}$
Given a, can you solve b?
For c, to get the chance that exactly two people got their own gifts, you have ${5\choose 2}$ ways to get those people and $2$ derangements of the remaining gifts, for $20$ ways to do this and a chance of $\frac {20}{120}=\frac 16$ You should be able to do the others.
d What is the probability that exactly four people received their own gifts?