## Homework 10, Due Thursday, 04/21

For each of these questions, you must show your work and explain your answer. You will have to write down some English sentences. Answers that consist of a single number will receive less than 50% credit!

When factorials are involved, leave your answer in terms of factorials (e.g., 5!/(3! ⋅ 2)).

1. Trip Planning. For a trip, you packed 2 pairs of sneakers, 3 pairs of jeans, 4 pairs of shorts, 5 shirts, 2 sweaters and 3 rings. How many different outfits can you make from the clothing you packed, if an "outfit" consists of 1 pair of sneakers, 1 pair of jeans or shorts (but not both), 1 shirt, at most 1 sweater and any number of rings? (Note: wearing rings on different fingers does not count as a different "outfit.")

Note: Assume that your sneakers, jeans, shorts, shirts, sweaters and rings are distinguishable.

2. Sock Drawer. Suppose your sock drawer has 17 pairs of socks that are black, white or tan. Which of the following statements must be true? Justify your answer.

1. There are at least 5 pairs of black socks, at least 5 pairs of white socks and at least 5 pairs of tan socks.

2. There are at most 4 pairs of black socks, at most 4 pairs of white socks or at most 4 pairs of tan socks.

3. There are at least 6 pairs of black socks, at least 6 pairs of white socks or at least 6 pairs of tan socks.

3. Another Sock Drawer. Suppose that your sock drawer has 6 pairs of black socks, 5 pairs of white socks and 6 pairs of tan socks. How many different ways are there to pack 5 pairs of socks? You can bring as many pairs of socks of each color as you want. Assume that socks of the same color are not distinguishable.

4. Hotel Room Closet. In the hotel room, you hang up in the closet the 5 shirts and 2 sweaters that you packed. How many ways can you arrange the shirts and sweaters in the closet (from left to right) so that the 2 sweaters are adjacent to each other? (As before, assume that your shirts and sweaters are distinguishable.)