The vowels always occupy even places?

Question

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

The vowels always occupy even places?

Answer 1

Given : the word is ‘GANESHPURI.’ 

To find : number of arrangements so that the vowels occupy only even positions 

Number of vowels in the word ‘GANESHPURI’ = 4(A, E, I, U) 

Number of consonants = 6(G, N, S, H, R, I) 

Let a vowel be denoted by V Even positions are 2, 4, 6, 8 or 10 

Now, 

Fix the position by Vowels like this :

V

V

V

V

V

Now, 

Arrange 4 vowels at these 5 places Formula used: 

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

P(n, r) = \(\frac{n!|}{(n-r)!}\) 

Total number of arrangements of vowels 

= the number of arrangements of 5 things taken 4 at a time 

= P(5, 4)

= \(\frac{5!|}{(5-4)!}\)

= \(\frac{5!|}{1!}\)

= 5! 

= 5 × 4 × 3 × 2 × 1 

= 120 

The remaining 1 even place and 5 odd places can be occupied by 6 consonants 

So, arrange 6 consonants at these remaining 6 places 

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 of consonants 

= the number of arrangements of 6 things taken all at a time 

= P(6, 6)

= \(\frac{6!}{(6-6)!}\)

= \(\frac{6!}{0!}\)

{∵ 0! = 1} = 6! = 6 × 5 × 4 × 3 × 2 × 1 

= 720 

Hence,

Number of arrangements so that the vowels occupy only even positions = 120 × 720 

= 86400.