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...

1)

1. 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 nd 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 ckle. He changes his mind about the number of people he wants to invite. Suppose there are

ve 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 condent in his own gift. What is the probability that Tom

ends up receiving Finn's gift, but Finn does not receive Tom's gift?

2)

1. Suppose you are given a test composed of ten problems. Not all problems on the test carry the same

weight. The rst eight problems are worth as much as each other and in total sum to sixty points.

The last two problems are also worth as much as each other and in total sum to forty points. Each

problem is a TRUE/FALSE problem. In order to pass, you need to score at least sixty points on the test.

You forgot to study for the test. You decide to guess the answer to each problem. Your strategy

is to not answer false for any two consecutive questions. In how many ways can you answer all ten

questions on the test?

3

2. Your friend Nathan also forgot to study and decides to just guess the answer to each problem.

Given that Nathan is able to correctly guess the answer to the last two problems, what is the probability

that he passes the test?

3. Find the probability that Nathan passes the test. [Hint: First condition on the number of correctly

answered questions among the last two problems.]