How to use product rule to prove a finite set S's subset is $2^n$

Question

Use the product rule to show that the number of different subsets of a finite set S is $2^n$ where n is the number of a object of S.

Attempt:

In my opinion, product rule is a form like there are m choices and each choice has n choices, and using product rule can get there are $m\times n$ total choices. But how it works on counting the subset of a finite set?

Answer 1

Hint:

Let $|S|=n$. Then, for each of the $n$ items, you have exactly $2$ choices: to include it, or not to include it in the subset that you are forming.

Now, you can use the product rule as you mentioned.

Answer 2

Let $\{a_1,a_2,\cdots,a_n\}$ be given set. Look at the product $*=(1+a_1)(1+a_2)\cdots(1+a_n)$. When we expand this to see $$1+(a_1+a_2\cdots+a_n)+(a_1a_2+a_1a_3+\cdots)$$ Each term is associated to a set. The 1 is associated to empty set. The term that has product of all elements represent entire set. So the total number of terms is total number of subsets. But the total number of terms in the expansion is $2^n$. Because * is multiplication of $n$ binomials.