How many words can be formed out of the letters of the word, ‘ORIENTAL,’ so that the vowels always occupy the odd places?

Question

Philoid · Accepted Answer

Given: the word is ‘ORIENTAL.’

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

Number of vowels in the word ‘ORIENTAL’ = 4(O, I, E, A)

Number of consonants in given word = 4(R, N, T, L)

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 4 vowels at 4 places

Formula used:

Number of arrangements of n things taken all at a time = P(n, n)

∴ Total number of arrangements of vowels

= the number of arrangements of 4 things taken all at a time

= P(4, 4)

{∵ 0! = 1}

= 4!

= 4 × 3 × 2 × 1

= 24

The remaining 4 even places can be occupied by 4 consonants

So, arrange 4 consonants at remaining places