Maths Olympiad Prep

Library / /1 of 4

Combinatorics Difficulty 4.4 AIME Find the answer United States

Problem:

In how many ways can you rearrange the letters of "HMMTHMMT" such that the consecutive substring "HMMT" does not appear?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

There are 8!/(4!2!2!)=4208!/(4!2!2!)=420 ways to order the letters. If the permuted letters contain "HMMT", there are 5⋅4!/2!=605 \cdot 4!/ 2!=60 ways to order the other letters, so we subtract these. However, we have subtracted "HMMTHMMT" twice, so we add it back once to obtain 361 possibilities.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.