# Thread: simple permutations

1. ## simple permutations

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?

THANKS!!!

2. Originally Posted by differentiate
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?

THANKS!!!
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.