Consider the 3-lists taken from [3]. How many are there in which each element of [3] appears at least once?

I get $\displaystyle _{3}P_{3}$.

Answer the same question, but for 4-lists and 5-lists taken from [3].

For the 4-list I get $\displaystyle _{3}P_{3}$*3.

For the 5-list I get $\displaystyle _{3}P_{3}$*$\displaystyle 3^2$.

Are these answers correct?