# 1. Solve a recurrence

Let T(n) be defined by the recurrence

- T(0) = 1.
- T(1) = 0.
- T(2) = 7.
- For n≥3, T(n) = 6 T(n-3) + 7 T(n-2).

Give a closed-form expression for T(n).

# 2. Triplets

Suppose you have a collection of objects of varying weights, and there are exactly ways to make a sequence of three of these objects with total weight n. How many objects are there with weight n?

# 3. Birthdays

Suppose that P and Q each have birthdays that are equally likely to be any of the 365 days of the year,^{1} and that the two birthdays are independent. What is the probability that both birthdays occur on the same day of the week in a non-leap year?

We are conditioning on the event that neither of them is born on February 29th. (1)