Is there room at an interview for someone to silently figure something out...?
Frankly? No.
Not for three minutes, anyway.
That's why either (a) the permutations question is too hard, or (b) the interviewer cannot necessarily expect anyone to get the right answer, but should instead judge incidental aspects of the process. (This is what dkarl seems to be doing, and that's fine.)
If you are asked a hard question, I think it's better to steer on the side of "going down a lot of wrong alleys, out loud". Just be honest about it. If you find yourself lost, go ahead and say "oops, thinking out loud during interviews is not my strong suit and now I'm pretty sure I'm lost". Find your own bugs and laugh at them. Backtrack. Start over. Restate the problem. Sketch a unit test that describes the answer that you can't quite derive yet. Define a simpler problem that you can solve out loud, and solve that first. And, of course, always do the stupidest thing that could possibly work.
It occurs to me that the average student spends all kinds of time in school studying how to solve problems quietly and completely at one's desk, and very little time practicing the art of sketching half-baked solutions out loud while waving one's hands. But you can practice that.
Frankly? No.
Not for three minutes, anyway.
That's why either (a) the permutations question is too hard, or (b) the interviewer cannot necessarily expect anyone to get the right answer, but should instead judge incidental aspects of the process. (This is what dkarl seems to be doing, and that's fine.)
If you are asked a hard question, I think it's better to steer on the side of "going down a lot of wrong alleys, out loud". Just be honest about it. If you find yourself lost, go ahead and say "oops, thinking out loud during interviews is not my strong suit and now I'm pretty sure I'm lost". Find your own bugs and laugh at them. Backtrack. Start over. Restate the problem. Sketch a unit test that describes the answer that you can't quite derive yet. Define a simpler problem that you can solve out loud, and solve that first. And, of course, always do the stupidest thing that could possibly work.
It occurs to me that the average student spends all kinds of time in school studying how to solve problems quietly and completely at one's desk, and very little time practicing the art of sketching half-baked solutions out loud while waving one's hands. But you can practice that.