Question

If some or all of n things be taken at a time, prove that the number of combinations is $2^n-1$.

Answer 1

Each of the things can be taken or left out i.e. each gives two ways. So the combinations of n things will be
2 x 2 x 2........ n factors = $2^n$ But this includes the case when all have been left out. So the number of combinations is $2^n-1.$