Question

Find the total number of ways in which $30$ distinct objects can be put into two different boxes so that no box remains empty.

Answer 1

These are 30 objects,, and each object has 2

options i.e, either go in box A or in box 13.

Hence Total ways = $2\times 2\times ---30$ $times=2^{30}$

Now in these $2^{30}$ ways, these would be one

particular instance when all objects would have gone to box A.

Also, there be one particular instance when all object would have gone to box B.

so these are 2 instance when one of the box would have been empty.

$\Rightarrow$ Required number of ways $=2^{30}-2$