Question

In how many ways the word regulations can be arranged with the vowels together

Answer 1

There are (equally likely ways to choose the two positions that will be reserved for the letters (Note this does not count where as individual letters go, we are just putting reserved signs on the two positions.)

If the two positions have a gap of between them, there are places where the leftmost position can be, and then the other position is determined. It follows that our probability is.

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Answer 2

ways. So total ways where R and E have 4 letters between them = 6 * 2 * 9! You could take (94)×4! ×2×6!