24.4. (SFRY, 72). For each value of $n \in \mathbf{N}$, find the largest number $k \in \mathbf{N}$ with the following property: in a set consisting of $n$ elements, one can choose $k$ different subsets, any two of which have a non-empty intersection.
24.4. Let us fix the element $a_{1}$ of the set $X=\left\{a_{1} ; a_{2} ; \ldots ; a_{n}\right\}$ and consider only subsets containing $a_{1}$. The number of such subsets is equal to the number of subsets of the set $\left\{a_{2} ; \ldots ; a_{n}\right\}$, i.e., the number $2^{n-1}$. Therefore, $k \geqslant 2^{n-1}$. On the other hand, suppose more than $2^{n-1}$ subsets of the set $X$ are chosen. We will divide all possible subsets of the set $X$ into $2^{n-1}$ pairs, combining a subset and its complement into a pair. Then, by the pigeonhole principle (Theorem 1), at least two of the chosen subsets must form a pair and, consequently, do not intersect. Thus, $k=2^{n-1}$.