the number of ways in which 6 different balls can be put in two boxes of different sizes such that no box is empty is

You may proceed this way :

There are 2 distinct boxes (say A and B) and n distinct objects.

So each of n objects can go to either of the two boxes.

So total no. of possibilities = 2^n

Now the problem is, these 2^n cases contain two cases where:

# all objects are in box A

# all objects are in box B

These two cases make a box empty, which is prohibited. So we need to exclude these 2 cases.

So total no. of ways =( 2^n) - 2

But here n = 6

Hence answer = 2^6 - 2 = 64 - 2 ,= 62

Regards

Arun (askIITians forum expert)