AMC 12 Mock Exam 4 Problem 5

Treat Ana and Ben as a block (in $2$ internal orders): there are $2\cdot4!=48$ lineups with them adjacent. Among these, count those with Cal at an end. If Cal is in position $1$, the block occupies one of the $3$ adjacent pairs among positions $2$ through $5$, in $2$ orders, and the other two friends fill the remaining spots in $2$ ways: $3\cdot2\cdot2=12$ lineups. The same holds with Cal in position $5$. So the answer is $48-24=\boxed{24}$.