If there are three ways of entering a house, in how many ways can 4 guests enter and leave the house.

For only one 1 person the choice is 3
ways.

For only 2 persons  (A and), AEach has 3 choices. For
any one ofthe way A goes , B has  3choices. So for each of 3 choices of , the other has
3 choices. So in all  for 2 persons, 3*3 = 9 ways are
there.

Like wise we can extend for number n persons and m
doors , they could exit in n^m ways

In this case n = 4 and
m= 3:

Since each of the 4 guests is independent to take any
of the 3 ways, irrespctive of what the other choses, the number of ways
is

3*3*3*3 = 3^4 = 81..