Question 1007116
in how many different ways can the letters of word "MATHEMATICS" 
be arranged so that the vowels always come together?
<pre>
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</pre>