Library / /596 of 740

, 2022

Combinatorics Difficulty 5.3 AIME, harder Prove it United States

Problem:

The English alphabet, which has 26 letters, is randomly permuted. Let $p_{1}$ be the probability that $\mathrm{AB}$ , $\mathrm{CD}$ , and $EF$ all appear as contiguous substrings. Let $p_{2}$ be the probability that $ABC$ and $DEF$ both appear as contiguous substrings. Compute $\frac{p_{1}}{p_{2}}$ .

Solution

Solution:

There are $23!$ ways to arrange the alphabet such that $AB$ , $CD$ , and $EF$ all appear as contiguous substrings: treat each of these pairs of letters as a single merged symbol, which leaves 23 symbols to permute. Similarly, there are $22!$ ways to arrange the alphabet such that $ABC$ and $DEF$ both appear as contiguous substrings. Thus, $p_{1} = 23! / 26!$ and $p_{2} = 22! / 26!$ , so the answer is $23! / 22! = 23$ .

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.