To find the total number of proper subsets of a set containing [tex]\( n \)[/tex] elements, let's start by understanding what a proper subset is. A proper subset is a subset that contains some or none of the elements of the original set but is not equal to the original set itself.
### Step-by-Step Solution:
1. Total Number of Subsets:
For a set with [tex]\( n \)[/tex] elements, the total number of subsets (including the empty set and the set itself) is given by [tex]\( 2^n \)[/tex]. This is because each element can either be included in a subset or not, giving [tex]\( 2 \)[/tex] choices per element and thus [tex]\( 2 \times 2 \times \cdots \times 2 = 2^n \)[/tex] total subsets.
2. Proper Subsets Count:
Proper subsets are all the subsets of the original set except the set itself. Thus, the total number of proper subsets is the total number of subsets minus one (the set itself):
[tex]\[
\text{Total number of proper subsets} = 2^n - 1
\][/tex]
Given the choices:
- [tex]\( 2^n \)[/tex]: This represents the total number of subsets, not just the proper ones.
- [tex]\( 2^{n-1} \)[/tex]: This does not relate directly to the subset counting formula.
- [tex]\( 2^n - 1 \)[/tex]: Matches our derived formula for the total number of proper subsets.
- [tex]\( 2(n-1) \)[/tex]: This is simply double the size of a slightly reduced version of the set's count, which is irrelevant here.
- [tex]\( n \)[/tex]: This suggests only [tex]\( n \)[/tex] subsets, which is incorrect.
- [tex]\( 2 n \)[/tex]: Again, does not relate directly to the subset count.
Therefore, the correct answer is:
[tex]\[ 2^n - 1. \][/tex]
So, if asked what the total number of proper subsets of a set containing [tex]\( n \)[/tex] elements is, you would choose:
[tex]\[ \boxed{2^n - 1} \][/tex]
