Counting permutations of students standing in line Say I have k students, four of them are Jack, Liz, Jenna and Tracy. I want to count the number of permutations in which Liz is standing left to Jack and Jenna is standing right to Tracy. I define $A = ${Liz is left to Jack} so $|A| = \frac{k!}{2}$. The same goes for $B$ with Jenna and Tracy. 
I know that $$|A \cap B| = |A| + |B| - |A \cup B|$$
But how do I find the union? I'm guessing it involves inclusion-exclusion, but I can't remember how exactly. 
Any ideas? Thanks!

The order relationship between Liz and Jack is independent of that between Jenna and Tracy. You already know that there are $k!/2$ permutations in which Liz stands to the left of Jack. In each of those Jenna can be on the right of Tracy or on her left without affecting the order of Liz and Jack, so exactly half of these $k!/2$ permutations have Jenna to the right of Tracy. The answer is therefore $k!/4$.
