Q22 of 140 Page 296

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


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.


More from this chapter

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20

In how many ways can the letters of the word ‘FAILURE’ be arranged so that the consonants may occupy only odd positions?

 21

In how many arrangements of the word ‘GOLDEN’ will the vowels never occur together?

 23

How many permutations can be formed by the letters of the word ‘VOWELS’, when

(i) there is no restriction on letters;


(ii) each word begins with E;


(iii) each word begins with O and ends with L;


(iv) all vowels come together;


(v) all consonants come together?


 24

How many numbers divisible by 5 and lying between 3000 and 4000 can be formed by using the digits 3, 4, 5, 6, 7, 8 when no digit is repeated in any such number?