In how many ways can the letters of the word MACHINE be arranged so that the vowels occupy only the odd positions?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 576
Place the vowels first in the allowed positions, then fill the remaining positions with the consonants. The two steps are independent, so their counts are multiplied.
MACHINE has 7 different letters: vowels A, I, E (3) and consonants M, C, H, N (4)
Odd positions among 7: 1st, 3rd, 5th, 7th, that is 4 places
Ways to place the 3 vowels in 4 odd places: \(^4P_3=4\times3\times2=24\)
The 4 consonants fill the 4 places left: \(4!=24\) ways
Total \(=24\times24=576\)