Words are formed by arranging the letters of the word "STRANGE" in all possible manner. Let $m$ be the number of words in which vowels do not come together and $'n'$ be the number of words in which vowels come together. Then find the ratio of $m : n$. (where $m$ and $n$ are coprime natural number).

The word STRANGE has 7 letters total, so it has 7 factorial, or 5040, total arrangements. The 2 vowels can be treated as a single unit, creating 6 units that arrange in 6 factorial ways, and the vowels arrange in 2 factorial ways, yielding 720 multiplied by 2 for a total of 1440 arrangements where vowels are together (n). The number of arrangements where vowels do not come together (m) is 5040 minus 1440, which equals 3600. The ratio of m to n is 3600 to 1440, which simplifies to 15 to 16.

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