**Question:**

How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:

(i) the letter G always occupies the first place?

(ii) the letters P and I respectively occupy first and last place?

(iii) the vowels are always together?

(iv) the vowels always occupy even places?

**Solution:**

The word GANESHPURI consists of 10 distinct letters.

Number of letters = 10!

(i) If we fix the first letter as G, the remaining 9 letters can be arranged in 9! ways to form the words.

∴ Number of words starting with the letter G = 9!

(ii) If we fix the first letter as P and the last letter as I, the remaining 8 letters can be arranged in 8! ways to form the words.

∴ Number of words that start with P and end with I = 8

(iii) The word GANESHPURI consists of 4 vowels. If we keep all the vowels together, we have to consider them as a single entity.

So, we are left with the remaining 6 consonants and all the vowels that are taken together as a single entity. This gives us a total of 7 entities that can be arranged in 7! ways.

Also, the 4 vowels can be arranged in 4! ways amongst themselves.

By fundamental principle of counting:

Total number of arrangements $=7 ! \times 4 !$ words.

(iv) The word GANESHPURI consists of 4 vowels that have to be arranged in the 5 even places. This can be done in 5! ways.

Now, the remaining 6 consonants can be arranged in the remaining 6 places in 6! ways.

Total number of words in which the vowels occupy even places $=5 ! \times 6 !$