Competition problem — Combinatorics

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Combinatorics Difficulty 5.2 AIME, harder Prove it Philippines

Problem:

The letters of the word MATHEMATICS are rearranged to form distinct strings of the same 11 letters. What proportion of these strings do not contain the string MATH?

Solution

Solution:

By treating the string MATH as a single character, we find that there are exactly eight distinct "letters" to rearrange. Hence, there are $8!$ such words out of a total of $\frac{11!}{2!2!2!}$ . This makes for a proportion of $\frac{8! \cdot 8}{11!} = \frac{4}{495}$ , and so $\frac{491}{495}$ of the words do not contain the aforementioned string.

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