157. Ballot Papers. A physical society needed to hold elections for three leadership positions. For each of the three positions, there were 3, 4, and 5 candidates, respectively. To ensure that the number under which each candidate is listed on the ballot does not influence the voting results, it was decided to apply a rule that in the list of candidates for each position, each candidate must appear under each number an equal number of times. What is the minimum number of different ballots required to comply with this rule?
157. Perhaps someone will immediately say that it is necessary to take $3 \cdot 4 \cdot 5=60$ different ballots. However, if we add two fictitious names to the group of three candidates and one fictitious name to the group of four candidates, then only 5 different ballots will be required. This technique not only reduces printing costs but also allows for the collection of statistical[^31] material, which can help determine what has a greater influence on the voting results: the number under which a candidate is listed, or their surname.

[М. S. Klamkin, M. M., 30, 110 (November 1956).]