  In this paper we consider mixed volumes of combinations of hypersimplices.
These numbers, called "mixed Eulerian numbers", were first considered by A.
Postnikov and were shown to satisfy many properties related to Eulerian
numbers, Catalan numbers, binomial coefficients, etc. We give a general
combinatorial interpretation for mixed Eulerian numbers and prove the above
properties combinatorially. In particular, we show that each mixed Eulerian
number enumerates a certain set of permutations in $S_n$. We also prove several
new properties of mixed Eulerian numbers using our methods. Finally, we
consider a type $B$ analogue of mixed Eulerian numbers and give an analogous
combinatorial interpretation for these numbers.
