The number of subsets of a 5-element set is 2^5 = 32. One of those is improper, so the number of proper subsets is 31.

The number of subsets of an n-element set is 2^n. One of them is improper. One of them is empty.

Consider a 3-element set, {a, b, c}. It will have 2^3 = 8 subsets. Now consider counting in base-2 (binary). This is what counting from 0 to 7 looks like:

000

001

010

011

100

101

110

111

If you let the digits correspond to a, b, c, and you put the corresponding element of the set into the subset when the digit is 1, you have

000 corresponds to { }

001 corresponds to {c}

010 corresponds to {b}

011 corresponds to {b, c}

and so on. The last subset, {a, b, c} is improper because it contains all of the elements of the original set.

As you can see, there is a 1 to 1 correspondence between binary number from 0 to 2^n-1 and subsets of an n-element set. Thus, there are 2^n subsets of an n-element set.

The number of subsets of an n-element set is 2^n. One of them is improper. One of them is empty.

Consider a 3-element set, {a, b, c}. It will have 2^3 = 8 subsets. Now consider counting in base-2 (binary). This is what counting from 0 to 7 looks like:

000

001

010

011

100

101

110

111

If you let the digits correspond to a, b, c, and you put the corresponding element of the set into the subset when the digit is 1, you have

000 corresponds to { }

001 corresponds to {c}

010 corresponds to {b}

011 corresponds to {b, c}

and so on. The last subset, {a, b, c} is improper because it contains all of the elements of the original set.

As you can see, there is a 1 to 1 correspondence between binary number from 0 to 2^n-1 and subsets of an n-element set. Thus, there are 2^n subsets of an n-element set.