Question

In how many ways can the letters of the word ‘STRANGE’ be arranged so that
(i) the vowels come together?

(ii) the vowels never come together?

Answer 1

Given as

The word ‘STRANGE’

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

(i) vowels come together?

Considering as 2 vowels as one letter therefore we will have 6 letters which can be arranged in ⁶P₆ ways.

(A,E) can be put together in ²P₂ ways.

Thus, the required number of words are

On using the formula,

P (n, r) = n!/(n – r)!

P (6, 6) × P (2, 2) = 6!/(6 – 6)! × 2!/(2 – 2)!

= 6! × 2!

= 6 × 5 × 4 × 3 × 2 × 1 × 2 × 1

= 720 × 2

= 1440

Thus, total number of arrangements in which vowels come together is 1440.

(ii) vowels never come together?

Total number of letters in the word ‘STRANGE’ is ⁷P₇ = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

Therefore,

The total number of words in which vowels never come together = total number of words – number of words in which vowels are always together

= 5040 – 1440

= 3600

Thus, the total number of arrangements in which vowel never come together is 3600.