Given: the word is ‘STRANGE.’

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

Number of vowels in word ‘STRANGE’ = 2(E, A)

Number of consonants = 5(S, T, R, N, G)

Let vowels be denoted by V

Odd positions are 1, 3, 5 or 7

So, fix the position by Vowels like this:

Now, arrange these 2 vowels at 4 odd 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 4 things taken 2 at a time

= P(4, 2)

= 4 × 3

= 12

The remaining 3 even places and 2 odd places can be occupied by 5 consonants

So, arrange these consonants at these 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 5 things taken all at a time

= P(5, 5)

∵ 0! = 1}

= 5!

= 5 × 4 × 3 × 2 × 1

= 120

Hence, the number of arrangements so that the vowels occupy only odd positions = 12 × 120 = 1440