in how many different ways can the letters of word "MATHEMATICS"
be arranged so that the vowels always come together?
MATHEMATICS
1. There are 4 vowels, A,A,E,I, of which there is one pair, the
A's, that are indistinguishable.
2. There are 7 consonants C,H,M,M,S,T,T, of which there are two
pairs, the M's and the T's, of indistinguishable ones.
3. If we let V stand for the vowels and C for the consonants,
we have these 8 basic cases where the vowels come together.
VVVVCCCCCCC
CVVVVCCCCCC
CCVVVVCCCCC
CCCVVVVCCCC
CCCCVVVVCCC
CCCCCVVVVCC
CCCCCCVVVVC
CCCCCCCVVVV
1. The number of distinguishable ways to arrange the vowels
AAEI is 4!/2!.
2. The number of distinguishable ways to arrange the consonants
CHMMSTT is 7!/(2!2!).
3. There are 8 basic cases above where the vowels all come
together.
Answer: (4!/2!)[7!/(2!2!)](8) = 120960
Edwin
