Question

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

Answer 1

To find the number of ways in which the letters of the word ‘MACHINE’ can be arranged such that the vowel may occupy only odd positions. We will first find the number of vowels and consonants present in the word ‘MACHINE’. Then we will find the number of ways using the concept of permutation.

Complete step by step answer:
In the given word ‘MACHINE’, there are $7$ letters in which $3$ are vowels and $4$ are consonants. A, I and E are vowels and M, C, H and N are consonants. There are seven positions, out of which four positions are odd and three positions are even. Now, three vowels can be placed at any of the four positions. As we know, number of ways of arranging $r$ from as set of $n$ if order matters is given by,
$^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$

Number of ways of arranging the vowels ${ = ^4}{P_3} = \dfrac{{4!}}{{\left( {4 - 3} \right)!}}$
On simplifying, we get
Number of ways of arranging the vowels $= \dfrac{{4!}}{{1!}} = 24$
When we arrange three vowels at three places, we are left with four positions and four consonants which can be arranged in $^4{P_4}$ ways.
Number of ways of arranging the consonants ${ = ^4}{P_4} = \dfrac{{4!}}{{\left( {4 - 4} \right)!}}$
On simplifying, we get
Number of ways of arranging the vowels $= \dfrac{{4!}}{{0!}} = 24$
Required number of ways $= 24 \times 24 = 576$

Therefore, the number of ways in which the letters of the word ‘MACHINE’ can be arranged such that the vowel may occupy only odd positions is $576$.

Note: Here, the order did matter, so we have used the concept of permutation as permutation is an arrangement in a definite order. But, if we are not interested in order, then we have used combination because a combination is a grouping or subset of items where the order does not matter.