Question

A man is visiting a village where each person either always tells the truth or always lies. There are 5 villagers standing in a row, and the man asks each of them how many of the 5 men standing in the row always tell the truth. Each villager gives an integer answer from 0 to 5 (inclusive). How many possible multi-sets of answers could the man receive from the villagers?

Details and assumptions
A multi-set is a set in which the elements are allowed to be repeated, e.g. {1,1,2,2}. This distinction is made because a set, by definition, should contain distinct elements. The multi-sets {1,2,2} and {1,1,2} are distinct multi-sets, but come from the same set {1,2}.

Changing the order of the villagers' answers does not change the set of answers received. I.E. the multi-set {1,2,3,5,5} is the same as the set {5,3,2,1,5}.

Answer 1

Only a hint: do a case by case analysis. You need N observation tho