Hi everybody,

First of all, I'm sorry, I'm afraid this will be a long post. =P And now a disclaimer: I'm posting here a set of three problems, but before people come bashing on me saying that I should do my homework myself, I'll let it clear from now that you can help in as many problems as you want.

OK, so for the issue itself. These problems come from my last test, and thing is, you can solve them using recurrence relations, generating functions or inclusion exclusion principle (you can also solve via traditional combinatory reasoning, worth less points, obviously). For didatic purposes, I'm asking hints for all three methods in every question (so, this will - hopefully - be a long topic too =P).

So for the problems themselves:

1) During 6 days of the week (from monday to saturday), a young man spends two afternoons in front of the computer, two in front of the TV and two in front of the video game. He spends the whole afternoon in front of only one equipment. Considering he doesn't spend two consecutive afternoons in front of the same equipment, in how many different ways can he program the 6 afternoons of his week (i.e., attributing equipments to the 6 aternoons)?

2) A young man drinks two soda bottles in each one of the afternoons from monday to saturday. Supposing he buys 6 identical bottles of soda brand A and 6 identical bottles of soda brand B, determine the number of ways to distribute the 12 bottles in the 6 days (two a day).

3) Each soldier of a group of $\displaystyle n$ must get a bubble gum. There are three available brands of gum, A, B and C, and the number of distributed gums from brand A must be even. Determine the number of ways to make such a distribution.

PS: my title doesn't make any sense, please replace "Assortment" with "Assorted".

Thanks in advance,