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}.
Only a hint: do a case by case analysis. You need N observation tho
To find the number of possible multi-sets of answers that the man can receive from the villagers, we need to consider the two cases: when the villagers always tell the truth and when the villagers always lie.
Case 1: Villagers always tell the truth
In this case, each villager will give a number from 0 to 5 based on the number of villagers who always tell the truth. Since each villager can give any integer answer from 0 to 5, there are 6 choices for each villager. Therefore, the total number of possible multi-sets in this case would be 6^5.
Case 2: Villagers always lie
In this case, the villagers will intentionally give incorrect answers. Since each villager can still give any integer answer from 0 to 5, there are still 6 choices for each villager. Therefore, the total number of possible multi-sets in this case would also be 6^5.
However, we need to consider that the order of the villagers' answers doesn't matter, as mentioned in the given information. This means that if two multi-sets have the same elements but in a different order, they are considered the same multi-set.
To account for this, we note that there are multiple ways to order the answers within each multi-set. The number of ways to order the answers within a multi-set with k elements is k!. Therefore, for each multi-set, we need to divide the total number of arrangements by the number of ways to order the answers within that multi-set.
In this case, since each multi-set has 5 elements (since there are 5 villagers), we divide the total number of multi-sets by 5!.
Therefore, the total number of possible multi-sets that the man can receive from the villagers is (6^5) / (5!).
Calculating this, we get:
(6^5) / (5!) = (6 * 6 * 6 * 6 * 6) / (5 * 4 * 3 * 2 * 1) = 7776 / 120 = 64.8
Since we cannot have a fraction of a multi-set, we round down to the nearest whole number. Therefore, the man can receive a total of 64 possible multi-sets of answers from the villagers.