Q. 44.3( 3 Votes )

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

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


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


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