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

1.	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!× = \(\frac{4!}{(4-3)!}\times4\)! = \(\frac{4!}{1!}\times4!\) = \(\frac{24}{1}\times24\) = 576. Hence the total number of word in which vowel occupy odd positions only is 576.

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!× = \(\frac{4!}{(4-3)!}\times4\)!

= \(\frac{4!}{1!}\times4!\) = \(\frac{24}{1}\times24\) = 576.

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