Probability question: names in hats

[robvalue]

That was the very first answer I came up with when I tried this Sadly, it's wrong.

You're on the right track, but the probability for the second person depends on whether or not the first guy took his name already. If he took it, his probability is 8/9 again. If it didn't get taken, it's 7/8. Similarly, each pick will depend on whether his own one has been taken.

I can't get my head around how you account for all the combinations as they can all have different probabilities.