## Task 3 - 331213

In a beauty contest for poodles, Asta, Benno, Cäsar, and Dolly face a jury of four members. Each jury member votes for one of the dogs by raising a card with the initial letter of the dog's name. As a rule for evaluating this voting result, it was stipulated: If two dogs clearly receive more votes than the other two dogs, they are considered qualified. However, if the voting result does not clearly show two dogs with more votes than the other two, a supplementary regulation will be made (e.g., a new vote will be scheduled).

Determine the number of all those different voting results that lead to this latter case! Voting results are considered the same only if they not only match in the number of votes for the dogs but also in which jury members voted for the respective dogs. For example, the voting results $A A B C$ and $C A B A$ are considered different.
For each voting result, exactly one of the following five statements applies:

(1) A dog receives all four votes.

(2) A dog receives three votes, and a second dog receives one vote.

(3) Two dogs each receive two votes.

(4) A dog receives two votes, and two other dogs each receive one vote.

(5) Each dog receives one vote.

The case where not exactly two dogs receive a higher number of votes than the other two dogs occurs if and only if one of the statements (1), (4), (5) applies.

If (1) applies, the voting result is uniquely determined by specifying the dog that received four votes. Therefore, there are exactly 4 voting results with (1).

If (4) applies, the voting result is uniquely determined by:

(a) Specifying the two dogs that each received exactly one vote (there are exactly $\binom{4}{2}=6$ possibilities),

(b) Specifying the jury members who chose these dogs (exactly $4 \cdot 3=12$ possibilities for each),

(c) Specifying which of the two other dogs received two votes (exactly 2 possibilities for each).

Thus, there are exactly $6 \cdot 12 \cdot 2=144$ voting results with (4).

If (5) applies, the voting result is uniquely determined by specifying the assignment between the jury members and the dogs they chose (exactly 4! = 24 possibilities). Therefore, the case where additional rules are required occurs in exactly $4+144+24=172$ voting results.

Note: This high percentage (approximately $67.2\%$ of $4^{4}=256$ possible voting results) shows that the evaluation rule is not very effective.

Suggestion: Determine the desired number also using a computer program that "tries out" all voting results!

Adapted from $[5]$