In how many ways can the letters of the word 'STRANGE' be arranged so that
(i) the vowels come together?
(ii) the vowels never come together? and
(iii) the vowels occupy only the odd places?
(i) Number of vowels = 2
Number of consonants = 5
Considering the two vowels as a single entity, we are now to arrange 6 entities taken all at a time.
Total number of ways = 6!
Also, the two vowels can be mutually arranged amongst themselves in 2! ways.
By fundamental principle of counting:
Total number of words that can be formed $=6 ! \times 2 !=1440$
(ii) Total number of words that can be made with the letters of the word STRANGE = 7! = 5040
Number of words in which vowels always come together = 1440
∴ Number of words in which vowels do not come together = 5040
(iii) There are 7 letters in the word STRANGE.
We wish to find the total number of arrangements of these 7 letters so that the vowels occupy only odd positions.
There are 2 vowels and 4 odd positions.
These 2 vowels can be arranged in the 4 positions in $4 \times 3$ ways, i.e. 12 ways.
The remaining 5 consonants can be arranged in the remaining 5 positions in 5! ways.
By fundamental principle of counting:
Total number of arrangements $=12 \times 5 !=1440$