The number of ways in which the letters of the word MACHINE can be arranged such that the vowels may occupy only odd positions is

Question

The number of ways in which the letters of the word MACHINE can be arranged such that the vowels may occupy only odd positions is

Smart Achievers · Accepted Answer

(4)

Total letters in the word MACHINE = 7

Number of vowels $= 3, i.e., A, I, E$

Number of ways of arranging vowels $= {}^{4}P_{3} = 4 \times 3 \times 2 = 24$

Number of ways of arranging consonants $= {}^{4}P_{4} = 4! = 24$

$∴$ Required number of ways $= 24 \times 24 = 576$

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