Find the number of ways in which the letters of the word ‘MACHINE’

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

$P(4,3) 4 ! x=\frac{4 !}{(4-3) !} \times 4 !$

$=\frac{4 !}{1 !} \times 4 !=\frac{24}{1} \times 24=576$

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