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In how many ways can the letters of the word ‘STRANGE’ be arranged so that the vowels occupy only the odd places?

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Best answer

Given as

The word ‘STRANGE’

Here's 7 letters in the word ‘STRANGE’, which includes 2 vowels (A,E) and 5 consonants (S,T,R,N,G).

The vowels occupy only the odd places

Here are 7 letters in the word ‘STRANGE’. Out of these letters (A,E) are the vowels.

Here are 4 odd places in the word ‘STRANGE’. The two vowels can be arranged in 4P2 ways.

The remaining 5 consonants an be arranged among themselves in 5P5 ways.

Therefore, the total number of arrangements is

On using the formula,

P (n, r) = n!/(n – r)!

P (4, 2) × P (5, 5) = 4!/(4 – 2)! × 5!/(5 – 5)!

= 4!/2! × 5!

= (4 × 3 × 2!)/2! × 5!

= 4 × 3 × 5 × 4 × 3 × 2 × 1

= 12 × 120

= 1440

Thus, the number of arrangements therefore that the vowels occupy only odd positions is 1440.

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