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