It's obviously easier to lay the carpet without any obstructions so do it first if you can. Have the doors been fitted and any allowance made for the carpet? Fitters are used to dealing with doors but if they are tight then you might have to take them off anyway.
In some respects it is easier to hang the doors first, so you can paint them before the carpet is in. Your carpet supplier should be able to tell you how much to allow between the door base and the floor. I hung my doors, then ordered carpet. It worked out just fine, and I would do it that way round again.
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