Seems to me that by assuming N is infinite you change a problem in combinations into a problem in permutations. In the real world, as each person is seated, the number of available outcomes for the next person changes. The number of remaining seats decreases, but the chance that someone else is already in that person's seat increases. Try it for two people or three. Where N=2, the probability of at least one person getting the right seat is the same as the probability of everyone getting the right seat, equals 1/2. That's not 1-(1/2)^2.

What am I missing?