Answer
If a set has x members, the number of subsets is $2^x$.
Work Step by Step
A null set has only 1 subset, the null set itself.
A set with 1 member has twice that many subsets (2), including the null subset and the set itself.
Look at the following example. Each time a member is added to the set, the number of subsets doubles. The subsets include all the subsets of the smaller set, plus a copy of those subsets with the new member added to each.
{ } : { }
{1} : { }, {1}
{1,2} : { }, {1}, {2}, {1,2}
{1,2,3} : { }, {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}
A set with 0 members has $2^0=1$ subsets.
A set with 1 member has $2^1=2$ subsets.
A set with 2 members has $2^2=4$ subsets.
A set with 3 members has $2^3=8$ subsets.
A set with 6 members has $2^6=64$ subsets., etc.
Since the number of subsets doubles, the number of subsets can be expressed as a power of 2, where the exponent is the number of members in the set.
