Suppose we have a house (with finitely many rooms) in which every room has an even number of doors. Prove that the number of doors from the house to the outside world is also even.
This was one of my homework questions in a counting and probability course and I would like some help.
You can treat the outside as a kind of room ... a really big room if you want. Now: note that every door will be counted for exactly two rooms: the ones on either side of the door. So: if you add up the number of doors for all rooms, then that should always be even. Since the total number of doors for the rooms inside the house is even, that means that the number of doors for the on big 'outside' room should be even as well ... i.e. there are an even number of doors from the house to the outside.