Number of ways in which `6` distinct objects can be kept into two identical boxes so that no box remains empty is

Options

A. `31`
B. `32`
C. `63`
D. `64`

Answer

A

Explanation

`(a)` Initially consider the boxes `B_(1)` and `B_(2)` to be distinct.
Object `'a'` can be kept in either of the boxes in `2` ways, similarly for all other things `implies` Total ways `=2^(6)`
But this includes when all the things are in `B_(1)` or `B_(2)`
`implies` Number of ways `=2^(6)-2`
Since the boxes are identical, therefore,
`(2^(6)-2)/(2)=2^(5)-1=31`