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.