Question 731622
The number of subsets with {{{1}}} to {{{n-1}}} elements from a set with {{{n}}} elements is {{{2^n-2=2(2^(n-1)-1)}}. If you count the empty set and  the whole set of {{{n}}} elements, there are {{{2^n}}} subsets.
 
How many subsets of {1,2,3,4,5,6,7,8,9,10,11} contain the number 5?
One of those subsets will be {5}, with just one element.
If you remove the number 5 from each of the subsets containing 5, you would get all the subsets of {1,2,3,4,6,7,8,9,10,11} and there is {{{2^10-2}}} or {{{2^10}}} of those (counting the empty set and the whole 10-element {1,2,3,4,6,7,8,9,10,11} set
That is the number of subsets containing 5, counting {5} and  {1,2,3,4,5,6,7,8,9,10,11}.
 
How many subsets of {1,2,3,4,5,6,7,8,9,10,11} contain exactly three elements, one of which is 3?
Removing 3 from each of those subsets would give you all the subsets of {1,2,4,5,6,7,8,9,10,11} with exactly 2 elements and that is {{{10*9/2=5*9=45}}} subsets. There are several different combination symbols for that and you know which one you are expected to use.
 
How many subsets of {1,2,3,4,5,6,7,8,9,10,11} have 5 elements but contain neither 3 not 5?
All of those subsets can be made from {1,2,4,6,7,8,9,10,11} and there is
{{{9*8*7*6*5/2/3/4/5=126}}} of them.