How to simulate a martingale process problem in R?

100 people are watching a theater.At the end of the show all of them are visiting the vesting room in order to take their coats.The man working on the vesting room give back people's coat totally at random.The participants that they will pick the right coat leave.The other that have picked the wrong one, give back the coat and the man again randomly gives back the coat.The process ends when all the customers of the theater take back their right coat.

I want to simulate in R this martingale process in order to find the expected time that this process will end. But I don't know how .Any help ? Something like:

# 100 customers
x = seq(1,100,by=1);x
# random sample from x 
y = sample(x,100,replace=FALSE)
x==y
# for the next iteration exclude those how are TRUE and run it again until everyone is TRUE

The expected time is how many iterations where needed .