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 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.