Some comments from Jose

1. For Problem 2, I would like to point out that the numbers that are both squares and cubes are sixth powers. Some of you did not know this fact. It follows from the uniqueness part of factorization into primes theorem. Just look at the exponents in the prime factorization of n. If n is both a square and a cube, then all exponents must be divisible by two and three, so they are divisible by 6, and thus we got a sixth power. This is a standard fact in number theory, and I did not take any points away if you didn't justify this. However, I was expecting you to state in your solution which numbers are both squares and cubes.
2. In Problem 3, many of you forgot that repetitions have to be taken out when counting words. We had many similar type permutations counting questions in the past, and a mistake like that in this homework is unacceptable. I took lots of points for this mistake.
3. In Problem 4, I noticed that many of you defined sets V_A , V_B, V_C, V_D and wrote four times the same thing. To save space and time (yours and mine), you can write something like "Let V_A be the set of words that have no A's. Define V_B, V_C and V_D similarly." It is totally ok to do that.
4. A general comment: When you are writing inclusion-exclusion proofs you don't have to prove the inclusion-exclusion principle again. It is better to define your sets and just use the principle. That saves you time and makes your solution easier to read. I would suggest that you read the official solutions and the examples from the book carefully to see how solutions that use this principle are usually written.
5. Also, if you are going to use a theorem from the book, please state it clearly and explain how you are using it.