\section*{Task 2 - 131222}

Each of the 41 students in a class had to participate in exactly three track and field running events.

Each of these students had to start once on lanes 1, 2, and 3.

Student A claims that due to these rules alone, there must be at least seven students in the class whose sequence of starting lanes is the same.

Student B, after some thought, claims that there must be at least eight such students. Verify whether each of these claims is correct.
}

Application of the Pigeonhole Principle:

If \(n\) objects are distributed among \(k\) sets, where \(n, k>0\), then there is at least one set that contains at least \(\left\lceil\frac{n}{k}\right\rceil\) objects. (\(\left\lceil\frac{n}{k}\right\rceil\) denotes the smallest integer \(g\) such that \(g \geq \frac{n}{k}\).)

Here, the \(n\) objects are simply the 41 students, and the \(k\) sets are the 6 sets corresponding to the 3! permutations of the 3 starting lanes, i.e., \(n=41\) and \(k=6\). Therefore, there must indeed be at least \(7\left(=\left\lceil\frac{41}{6}\right\rceil\right)\) students for whom the order of the starting lanes is the same. For 8 students, however, this statement cannot be made, as the simple equation

\[
7+7+7+7+7+6=41
\]

shows, which gives a possible "allocation" of the 6 permutations of the starting lanes, where there are no 8 students with the same order of starting lanes.