In how many ways can 4 consonants and 3 vowels be arranged in a row (a) so that the 3 vowels are always together, (b) so that the first and the last places are occupied by consonants.
(a) consider 3 vowels as one package, 4 consonants as 4 packages. Arranging these 5 packages, we have 5! ways. Besides, 3 vowels arrange within one package in 3! ways.
So in total, we have 3!*5! ways to arrange as requested.
(b) there're C(4,1) alternatives for first place; then, there're C(3,1) alternatives for last place. We're left with 2 consonants and 3 vowels to place in between. There're 5! ways to arrange these 5 entities => The number of way for us to arrange as requested is C(4,1)*C(3,1) * 5!
Very nice. Thanks.