In how many ways can the letters of the word 'STRANGE' be arranged so that

Solution:

(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 -">−-1440 = 3600

(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 $=12 \times 5 !=1440$