Arun sharma

Question

25. Find the number of ways in which the letters of the word MACHINE can be arranged so that the
vowels may occupy only odd positions.
(a) 4! × 4! (b) 7P3 × 4!
(c) 7P4 × 3! (d) none of these

Comments

nice question .

Answer 1

option A..??

4!*4!

Comments

Yeah .... solve  it please

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MACHINE

positions are 1,2,3,4,5,6,7

vowels are A,E,I

out of 4 odd positions(1,3,5,7) we can select 3 odd positions for vowels by C(4,3)=4 ways.

now for particular selected position triple we can arrange vowels and consonents as 4!*3!..and

since there are 4 such triples total no.of ways are 4*4!*3!=4!*4!..

hope u got..

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Ok it says that restriction is only on vowels and consonants  can go to the  even or odd place too .

  . Hope I m getting correctly  .Thank you

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yeah restriction is only on vowels...

Answer 2

In word  MACHINE there are 3 vowels  {A,E,I} and 4 consonent {M,C,H,N}
positions are 1,2,3,4,5,6,7   Vowels may occuppy only odd positions (4 places {1,3,5,7})

3 vowels can be arranged in 4 odd positions in ⁴P₃ ways
4 consonants  can be arranged in the remaining 4 positions in 4! ways

Total number of ways in which the letters of the word MACHINE can be arranged so that the
vowels may occupy only odd positions = ⁴P_{3 *} 4! = 4! * 4! = 24 * 24 = 576