1. Tardigrade
  2. Question
  3. Mathematics
  4. The 120 permutations of M A H E S are arranged in dictionary order, as if each were an ordinary five-letter word. The last letter of the 86 text th word in the list is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The $120$ permutations of $M A H E S$ are arranged in dictionary order, as if each were an ordinary five-letter word. The last letter of the $86^{\text {th }}$ word in the list is

Permutations and Combinations

Solution:

The first $24=4$ ! words begin with $A$, the next $24$ begin with $E$ and the next $24$ begin with $H$. So the $86^{\text {th }}$ begins with $M$ and it is the $86-72=14^{\text {th }}$ such word. The first 6 words that begin with $M$ begin with $M A$ and the next $6$ begin with $M E$. So the desired word begins with $M H$ and it is the second such word. The first word that begins with $M H$ is MHAES, the second is MHASE. Thus, $E$ is the letter we seek.