SOLUTION: Suppose A= {a,b,c,d,e,f}. How many subsets does A have? How many proper subsets does A have? How many subsets does A have that include the element b?

When creating a subset, for each of the 6 elements, 
there are 2 choices to make:

Choice 1:  Include the element in the subset.
Choice 2:  Do not include the element.

For instance, in creating the subset {a,d,f}, 

for element "a", choice 1 was made,
for element "b", choice 2 was made,   
for element "c", choice 2 was made, 
for element "d", choice 1 was made, 
for element "e", choice 2 was made, 
for element "f", choice 1 was made.

So, when creating a subset, there are 2 choices 
to make for a.
For each of those choices there are 2 choices 
to make for b.
For each of those choices there are 2 choices 
to make for c.
For each of those choices there are 2 choices 
to make for d.
For each of those choices there are 2 choices 
to make for e.
For each of those choices there are 2 choices 
to make for f.

So there are (2)(2)(2)(2)(2)(2) = 26 = 64 subsets.

Only 63, because the "subset" in which choice 1 was made for
every element is the "subset" is {a,b,c,d,e,f} is not
properly "sub", since it is the whole set.  So we subtract
it from the list of subsets. 

[However, the subset in which choice 2 was made for every element
is the "empty set", and it IS considered to be a proper subset.
The empty subset is written {} or ⌀.]

That's the same number as the subsets of {a,c,d,e,f}.  Just
list them all and insert "b" in every one.

Using the same reasoning as for the first problem above,
there are (2)(2)(2)(2)(2) = 25 = 32 subsets.

[It seems suprising that the number of subsets of A that contain
"b" is the same as the number of subsets that DO NOT contain "b".]

Edwin