Chef and K Segments

Chef has a difficult problem to solve. He has $N$ segments $[l_1, r_1], [l_2, r_2], \dots, [l_N, r_N]$ on the $x$-axis. A $K$-subset of these $N$ segments consists of any $K$ of these segments. For each $K$-subset, consider the common intersection of all the $K$ segments, i.e. the set of points which belong to each of the $K$ segments. The intersection of segments is also a segment; the length of