Question

If a finite set $A$ has $m$ elements, then the number of non-empty proper subsets of $A$ is

• A. $2^m$
• B. $2^m-1$
• C. $2^{m-1}$
• D. $2(2^{m-1}-1)$

Answer 1

The correct option is D $2(2^{m-1}-1)$

Since the subsets must be proper and non-empty, the 2 subsets which cannot be included are where no elements are selected (non-empty) and where all are selected (proper).

Total no. of subsets $=2^m$

Required no. of subsets $=2^m-2$

$=2(2^{m-1}-1)$