There are 5 doors in a hostel lounge. Two are red colored and the others are green. In how many ways can a guy enter the room and leave the room...

Hello!

I'm sure that all the doors are distinguishable, i.e. "enter the first red door" and "enter the second red door" are different events.

Then we can 1) enter a red door, leave (through) a green door or 2) enter a green door, leave a red door. The quantities of path in this two options must be added.

There are `5-2=3` green doors. For the option 1, we can enter each of `2` red doors and for each of them can leave through any of `3` green doors, so there are `2*3=6` paths. For the option 2, we can enter each of `3` green doors, leave through any of `2` red doors, again `3*2=6` paths.

And there are `6+6=12 ` paths total. This is the answer.