Question

In how many ways $5$ different balls can be distributed into $3$ boxes so that no box remains empty?

Answer 1

Lets fill each box with one ball.

So we are now left with 2 balls.

Now , let the boxes be labelled as A B C.

After each of them contains 1 ball each.

Possible ways of placing the remaining 2 balls:

Either both are placed in A or B or C -> 3 ways

Either (one placed in A and other in B ) or (one placed in B and other in C ) or (one placed in A and other in C ). -> 3 ways

Total , 6 ways.