Q. 6 D5.0( 2 Votes )

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 :

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:



Now, arrange 4 vowels at these 5 places


Formula used:


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



Total number of arrangements of vowels


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


= P(5, 4)




= 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)



Total number of arrangements of consonants


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


= P(6, 6)




{ 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


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