Question

The number of subsets that can be created from the set {1, 2, 3} is:
3
6
7
8

Answer 1

\[\large\rm S=\{1,2,3\}\]
The cardinality of the power set tells us the total number of subsets that can be created.\[\large\rm |\mathcal P(S)|=2^n\]It's 2 raised to the power of the number of elements in your set.

We have three elements in our set, so,
\(\large\rm 2^3=?\)

Answer 2

As for why \(|\mathcal P(S)|=2^n\) if \(|S|=n\), that's because any subset can be made by choosing \(k\) elements from \(S\), and adding up all the ways \(k\) elements can be taken. Suppose \(S=\{1,2,\ldots,n\}\), then you have
\[\begin{align*}
\text{empty set}&\implies \binom n0\text{ way of choosing a zero-element set}\\
\{1\},\{2\},\ldots,\{n\}&\implies \binom n1\text{ ways of choosing a one-element set}\\
\{1,2\},\{1,3\},\ldots,\{1,n\},\\
\{2,3\},\{2,4\},\ldots,\{2,n\},\ldots,\\
\{n-2,n-1\},\{n-2,n\},\\
\{n-1,n\}&\implies \binom n2\text{ ways of choosing a two-element set}\\
&~~~~\vdots
\end{align*}\]and so on. Adding up all the ways means computing the sum
\[\sum_{k=0}^n\binom nk=\sum_{k=0}^n\binom nk 1^{n-k}1^k=(1+1)^n=2^n\]where the first second equality relies on the binomial theorem.