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.