A summer camp gathers fifteen participants. Each day, three of them are chosen to do the dishes. At the end of the stay, it is observed that no matter how two participants are chosen, there is exactly one day where they were both on duty. How many days did the summer camp last?
Let $n$ be the number of days of the camp, and let's count the total number of pairs of participants. Since there are 15 participants, there are
$\binom{15}{2}=105$ pairs of participants. Each day, 3 participants are chosen. Among them, we can form $\binom{3}{2}=3$ pairs. Since each pair appears exactly once during the stay, this means that the total number of pairs is equal to $3 n$. It follows that $n=105 / 3=35$.