How many words can be formed out of the letters of the word ‘ARTICLE,’ so that vowels occupy even places?
Given: the word is ‘ARTICLE.’
To find: number of arrangements so that the vowels occupy only even positions
Number of vowels in word ‘ARTICLE’ = 3(A, E, I)
Number of consonants = 4(R, T, C, L)
Let vowel be denoted by V
Even positions will be 2, 4 or 6
So, fix the position by Vowels like this:
Now, arrange 3 vowels at 3 places
Formula used:
Number of arrangements of n things taken r at a time = P(n, r)
∴ Total number of arrangements of vowels
= the number of arrangements of 3 things taken 3 at a time
= P(3, 3)
{∵ 0! = 1}
= 3!
= 3 × 2 × 1
= 6
The remaining 4 odd places can be occupied by 4 consonants
So, arrange 4 consonants at remaining 4 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 even positions = 6 × 24 = 144