Question

How many different words can be formed from the letters of the word GANESHPURI when: the vowels are always together?

Answer 1

Hint – In this particular question first find out the number of consonants and the number of vowels in the given word then consider all the vowels as one then count the number of different letters including the vowels as one so arrange these letters and arrange the vowels so use these concepts to reach the solution of the question.

Complete step-by-step solution -
Given word:
GANESHPURI
As we know the number of vowels present in the English alphabets is 5 which are given as (A, E, I, O and U).
So the number of vowels present in the given word = A, E, U and I so there are 4 vowels present in the given word.
Now the total letters present in the given word = 10.
So the number of consonants present in the given word = (10 – 4) = 6
Now consider all 4 vowels present in the given word as one letter so the total letters now present are (6 + 1) = 7.
Now arrange these letters = 7!
And the arrangement of the vowels are (4!)
So the total different words can be formed from the letters of the word GANESHPURI when: the vowels are always together is the multiplication of the above calculated values so we have,
So the total different words are =$7!\left( {4!} \right)$
Now simplify this we have,
So the total different words are =$7.6.5.4.3.2.1\left( {4.3.2.1} \right) = 120960$ words.
So this is the required answer.

Note – Whenever we face such types of questions the key concept we have to remember is that if there are n letters in any particular words then the number of different words form from these letters is given as (n!) so separate consonants and the vowels in the given word and according to given criteria arrange them as above and simplify we will get the required answer.