Is there an easy way to sum $\sum_{1 \leq i < j < k \leq n} 1$?

Question

I'd like to have a standard procedure to sum terms like $$\sum_{1\,\leq\, i\, <\, j\, <\, k\, \leq\, n} 1$$ without having to "telescope" the sum, beggining from the outermost one and simply summing up each partial sum obtained. More generally, how do I perform sums like $$\sum_{1\, \leq\, i_{1}\, <\, i_{2}\, <\, \cdots\, <\, i_{k}\, \leq\, n} 1?$$ I teach this stuff in a basic combinatorics class and it got pretty messy last time. Thanks a lot in advance!