Learning How To Count

How many multigraphs on 12 vertices with 8 edges?

How many multigraphs on 6 vertices with 3 edges?

How many different marked (simple) graphs can be on 4 vertices?

How many different marked graphs with 4 edges can be on 9 vertices?

In how many ways can n people be placed in 10 rooms?(rooms can be empty)

How many bit strings of length ten begin and end with 1?

In how many ways can 10 different balls be layed out on three identical chests? (Some boxes may remain empty)

How many bit strings of length 12 contain exactly three 1?

The college has 18 students specializing in mathematics and 325 in computer science. In how many ways can you choose two students so that one is a mathematician and the other is a computer scientist?

3 passengers got into the elevator of the 12-storey building. Everyone, independently of the others, with the same probability can go out on any (starting from the 2nd) floor. How many ways can passengers get off the elevator?