## If $S$ has iid uniform points, how are points in $S^k$ distributed?

Take a rectangle with positive volume $$R\subset \mathbb{R}^d$$ and form a set $$S$$ of $$n$$ points iid uniform over $$R$$. Take a Jordan-measurable region $$U\subset R^k.$$ What is $$E\left[\left|U\cap S^k\right|\right]$$?