Q. 85.0( 1 Vote )

# 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

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