First problem:

There are 6 ways to enter.  

For each of those 6 ways to enter there are 6 ways to leave,
since the first problem states that you can leave by any door.

So that's 6 times 6 or 36 ways.
  
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Second problem:

There are 6 ways to enter.

For each of those 6 ways to enter there are only 5 ways to leave,
since the second problem states that you must leave by a different door.

So that's 6 times 5 or 30 ways.

Edwin