Explanation of this theorem on combinations There is this theorem in my book in the chapter of permutations and combinations which states: The total number of combinations of n different things taken any number of them at a time is $2^n$.
Its proof is given as: Each thing may be disposed of in two ways-it may or may not be chosen. Therefore, the total number of ways of disposing of all the n things $=2 \times 2\times 2 \times ...\text{n times}=2^n$. Hence, the number of combination (selections)=$2^n$. 
I searched the internet for this theorem but could not find it. Can someone please explain the meaning of this theorem to me as well as its proof. What does "disposed" mean here?

Suppose you have some set $S=\{s_1,...,s_n\}$.
You're essentially trying to see how many subsets of $S$ there are (ie. the cardinality of the powerset of $S$).
For each subset $S_i$, assign it the ordered tuple $((x_i)_1,...,(x_i)_n)$ where $(x_i)_k=\begin{cases}0 & s_k\not\in S_i\\1 & s_k\in S_i \end{cases}$
Clearly, the ordered tuples are different for each subset and all possible tuples in $\{0, 1\}^n$ are assigned to some subset. Therefore counting the number of subsets is equivalent to counting the number of possible ordered tuples.
Each ordered tuple is a sequence of $0$s and $1$s and therefore counting the number of possible tuples is equivalent to counting the number of different length $n$ binary strings.
The answer is $2^n$.

What I've laid out here is essentially the idea which your book explains. Each item in the set can be 'on' or 'off' and this is like binary.
