Q. 223.5( 4 Votes )

Find the number of ways in which the letters of the word ‘MACHINE’ can be arranged such that the vowels may occupy only odd positions.

Answer :

To find: number of words


Condition: vowels occupy odd positions.


There are 7 letters in the word MACHINE out of which there are 3 vowels namely A C E.


There are 4 odd places in which 3 vowels are to be arranged which can be done P(4,3).


The rest letters can be arranged in 4! ways


Formula:


Number of permutations of n distinct objects among r different places, where repetition is not allowed, is


P(n,r) = n!/(n-r)!


Therefore, the total number of words is


P(4,3)4!× = ×4!


= ×4! = ×24= 576.


Hence the total number of word in which vowel occupy odd positions only is 576.


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