Are the vowels always together? Are the vowels always together?

Question

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

Are the vowels always together?

Answer 1

Given : The word is ‘GANESHPURI.’ 

To find : number of words in which vowels are always together 

Number of vowels in this word = 4(A, E, I, U) 

Now, 

Consider these four vowels as one entity(AEIU together as a single letter) and arrange these letters 

So, the total number of letters = 7(AEIU G N S H P R) 

Formula used : 

Number of arrangements of n things taken all at a time = P(n, n)

P(n, r) = \(\frac{n!}{(n-r)!}\)

∴ Total number of arrangements 

= the number of arrangements of 7 things taken all at a time 

= P(7, 7)

= \(\frac{7!}{(7-7)!}\)

= \(\frac{7!}{0!}\)

{∵ 0! = 1} 

= 7! 

= 7 × 6 × 5 × 4 × 3 × 2 × 1 

= 5040 

Now,

4 vowels which are together as a letter can be arranged in 4! (like EAIU or AEUI) 

= 4 × 3 × 2 × 1 

= 24 ways 

∴ Total number of arrangements in which vowels come together = 24 × 5040 

= 120960