How many words can be formed out of the letters of the word, ‘ORIENTAL,’ so that the vowels always occupy the odd places?
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
Formula used:
Number of arrangements of n things taken all at a time = P(n, n)
∴ Total number of arrangements of consonants
= the number of arrangements of 4 things taken all at a time
= P(4, 4)
{∵ 0! = 1}
= 4!
= 4 × 3 × 2 × 1
= 24
Hence, the number of arrangements so that the vowels occupy only odd positions = 24 × 24 = 576