It should be clear (by the pigeonhole principle) that the five must be generated no later than fifth in the sequence if each other number is to have been generated only once.

Then either the five is generated first, second, third, fourth, or fifth.

P(1) = 1/5

P(2) = (4/5)x(1/5)

P(3) = (4/5)x(3/5)x(1/5)

P(4) = (4/5)x(3/5)x(2/5)x(1/5)

P(5) = (4/5)x(3/5)x(2/5)x(1/5)x(1/5)

Just add up the probabilities from there.