Question

What is the number of subsets of a set with n elements, containing a given element (when element becomes fixed, part of every subset)?

Answer 1

It is the same as to find a number of subsets of a set with $n-1$ elements. Namely, we take a set without the element(a subset that contains $n-1$ elements) and construct its all possible subsets. Then, we add this element to every subset. It remains to solve the following task: find a number of subsets of the set that contains $n-1$ elements. We use the multiplication principle of combinatorics. For any element of the set there are two options: it belongs to the subset or not. We receive: $2^{n-1}$ subsets.

Answer: in case we fix an element of the set and consider all subsets that contain this element, we receive $2^{n-1}$ different subsets.