Tom plans on throwing a holiday party for his friends. At this party, he decides he wants to organize a secret Santa style gift exchange. His plan is...

Tom plans on throwing a holiday party for his friends. At this party, he decides he wants to organize a secret Santa style gift exchange. His plan is all follows: the gifts will be put into identical boxes (same color, same size, completely indistinguishable from one another) so no one is able to tell whose gift is in a box. The boxes in turn will be randomly distributed to each guest. Ideally, Tom would like for no one to end up with his/her own gift. Tom wants to find out what the probability is that no one is assigned his/her own gift after the exchange. Suppose there are four people at his party (including himself). In how many ways can the boxes be distributed so that exactly one person is assigned his/her own gift?

2. Tom is fickle. He changes his mind about the number of people he wants to invite. Suppose there are five people total now (including himself). In how many ways can the boxes be distributed so that no one is assigned his/her own gift? Do not use explicit enumeration in this question. Instead, introduce the number of ways, Ni , of having i people getting their own gift. What is P5 i=0 Ni?

3. If the boxes are randomly distributed in a party of 5, what is the probability that no one is assigned his/her own gift?

4. Tom knows that his best friend Finn, one of the other 5 people attending the party, gives exceptionally good gifts. However, Tom does not feel as confident in his own gift. What is the probability that Tom ends up receiving Finn0 s gift, but Finn does not receive Tom0 s gift?