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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