Liar, Liar, Pants on Fire!

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 \( \{ 5, 3, 2, 1, 5\} \).