Question 15
Let $S=\{x_{1}, x_{2}, \ldots, x_{k+\ell}\}$ be a $(k+\ell)$-element set of real numbers contained in the interval $[0,1]$; $k$ and $\ell$ are positive integers. A $k$-element subset $A \subset S$ is called nice if
$$
\left|\frac{1}{k} \sum_{x_{i} \in A} x_{i}-\frac{1}{\ell} \sum_{x_{j} \in S \backslash A} x_{j}\right| \leq \frac{k+\ell}{2 k \ell} .
$$
Prove that the number of nice subsets is at least $\frac{2}{k+\ell}\binom{k+\ell}{k}$.
$$
\left|\frac{1}{k} \sum_{x_{i} \in A} x_{i}-\frac{1}{\ell} \sum_{x_{j} \in S \backslash A} x_{j}\right| \leq \frac{k+\ell}{2 k \ell} .
$$
Prove that the number of nice subsets is at least $\frac{2}{k+\ell}\binom{k+\ell}{k}$.