The letters of the word MOBILE are arranged at random in a row. What is the probability that the vowels occupy only the odd-numbered positions?

MOBILE has 6 distinct letters — vowels O, I, E and consonants M, B, L — so total arrangements = 6! = 720. There are exactly 3 odd positions (1st, 3rd, 5th); putting the 3 vowels there in 3! ways and the consonants elsewhere in 3! ways gives $6\times 6=36$ favourable arrangements. Probability = 36/720 = 1/20.