From the word, FORMULA, five letters are selected at random and arranged in a straight line. What is the probability that if one such arrangement is selected at random that the three vowels are together?
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There are 7! arrangements of FORMULA
If the 3 vowels are together, take them as a group.
There are 5! arrangements of the group with the other 4 letters.
However the group itself can be arranged in 3! ways.
That's 3!4! arrangements with the vowels together.
If 5 of the 7 are chosen, there are $\displaystyle 7p_5$ arrangements,
and there are 3! arrangements with the 3 taken as a group,
giving 3!3! arrangements that contain all 3 vowels.
The probability can be calculated then.