[Math] the number of subsets of a set with $n$ elements, containing a given element

Question

All of the subsets can be of most $n$ elements. One term, $a_1$ of our set, has to be in our subset. From there we can choose wether we want $a_2$ to be in our subset of not, that gives $2$ choices. We can choose if $a_3$ is in our subset or not, again two choices….We can choose if $a_n$ is in our subset or not. By the multiplication principle we have:

$$(1)(2^{n-1})=2^{n-1}$$

Possible subsets.

Is my answer/approach correct?

Answer 1

Your answer is perfectly correct. Here is a follow up :

Questions :

• What is the number of subsets in a set $E$ with n elements, containing a given $k\leq n$ elements?

• What is the number of subsets in a set $E$ with n elements,containing either $a_1$ or $a_2$ ($a_1$ and $a_2$ two given elements in $E$)?