Permutations Question

In how many ways can 4 consonants and 3 vowels be arranged in a row, if:

i) So that the 3 vowels are always together?

Show full working out please.

First permute the 4 consonants in one of 4! = 24 ways.

x C x C x C x C x

There are now 5 places to place the block of vowels,
indicated by x's in the string above,
which themselves can be permuted in 3! = 6 ways.

So overall, you have 5 * 4! * 3! ways = 5! * 3! = 6! (by coincidence) = 720

Another way to look at it:

1 block of vowels + 4 consonants = 5 objects with 5! permutations,
times 3! permutations for the vowels = 3! * 5! = 720.

treat the vowels as a "block" of 3 that is only internally permutable
the "block' can be internally permuted in 3! ways
and permuted along with the 4 consonants in 5! ways
ans: 3!*5! = 720
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a typo made 5! into 4!. that has been corrected and the solution left, as i feel it should be the preferred way of solving for this q.

probability of 1st vowel = 3/7

second vowel = 2/6

third vower = 1/5

total pro = 3/7*2/6*1/5

1 in 35