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.
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
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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