The $120$ permutations of $AHSME$ are arranged in dictionary order as if each were an ordinary five-letter word. 
The last letter of the $86$th word in this list is:
$\textbf{(A)}\ \text{A} \qquad \textbf{(B)}\ \text{H} \qquad \textbf{(C)}\ \text{S} \qquad \textbf{(D)}\ \text{M}\qquad \textbf{(E)}\ \text{E}$
We could list out all of the possible combinations in dictionary order.
$73rd:$ MAEHS                   
$74th:$ MAESH
$75th:$ MAHES
$76th:$ MAHSE
$77th:$ MASEH
$78th:$ MASHE
$79th:$ MEAHS
$80th:$ MEASH
$81th:$ MEHAS
$82th:$ MEHSA
$83th:$ MESAH
$84th:$ MESHA
$85th:$ MHAES
$86th:$ MHASE
We find that the $86th$ combination ends with the letter E. 
So the answer is $\textbf{(E)}\ E$.