SOLUTION: From 5 consonants and 4 vowels, how many words can be formed (with or without meaning) using 3 consonants and 2 vowels such that vowels are always together?

Suppose B,C,D,F,G are the consonants and A,E,I,O are the vowels

We choose the 3 consonants 5C3 = 10 ways.
We choose the 2 vowels 4C2 = 6 ways.

That's (10)(6) = 60 ways to choose what letters to use.

Suppose you chose B,D,G for the consonants and A,E for the vowels.

Then you can put the vowels together in either of 2 ways, (AE) or (EA).

So there are (60)(2) = 120 ways you can choose the letters and put the vowels together.

Suppose you chose (EA)

Then you have only 4 things to arrange, B,D,G and (EA).

You can choose the one to go first any of 4 ways.
Then you can choose the one to go second any of the remaining 3 ways.
Then you can choose the one to go third either of the remaining 2 ways.
Finally there is only 1 way to choose the only remaining way.

Answer: (120)(4)(3)(2)(1) or 2880 ways.

Edwin