In how many ways can the letters of the word PLANTED be arranged so that the vowels occupy only the odd positions?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 1440
Place the vowels first in the odd positions, then fill the remaining positions with the consonants. The two counts are multiplied.
PLANTED has 7 different letters: vowels A and E; consonants P, L, N, T and D
Odd positions: 1st, 3rd, 5th and 7th, that is 4 positions
The 2 vowels take 2 of the 4 odd positions: \(^4P_2=4\times3=12\) ways
The 5 consonants fill the 5 places left: \(5!=120\) ways
Arrangements \(=12\times120=1440\)
Why the other options are incorrect
Option (1) 240 — Incorrect. This is \(2\times5!\); it counts the 2 orders of A and E but never chooses which 2 of the 4 odd positions they take.
Option (2) 720 — Incorrect. This is \(^4C_2\times5!\); it chooses the places for the vowels but does not count A and E in 2 orders.
Option (4) 2880 — Incorrect. This is \(2\times1440\); the order of A and E is already counted in \(^4P_2\).