Question

The number of ways in which the letters of the word 'RUSSIAN' be arranged so that the consonants always occupy odd places, is

Answer 1

72

Answer 2

The word 'RUSSIAN' has 7 letters: 3 vowels (U, I, A) and 4 consonants (R, S, S, N). There are 7 positions in total: 4 odd positions (1, 3, 5, 7) and 3 even positions (2, 4, 6). The condition is that consonants must occupy odd places. Thus, all 4 consonants must be placed in the 4 odd positions. The number of ways to arrange the 4 consonants (R, S, S, N) in the 4 odd positions, accounting for the repeated 'S', is $\frac{4!}{2!} = 12$. The remaining 3 vowels (U, I, A) must occupy the 3 even positions. The number of ways to arrange these 3 distinct vowels in the 3 even positions is $3! = 6$. The total number of arrangements is the product of these two numbers: $12 \times 6 = 72$.