From a set of [tex]n[/tex] elements, there is only one subset that can be chosen that contains no elements (the empty set), i.e. [tex]\dbinom n0[/tex].
([tex]\dbinom nk=\dfrac{n!}{k!(n-k)!}[/tex] denotes the binomial coefficient)
Similarly, there are [tex]\dbinom n1[/tex] possible subsets consisting of only one element that can be taken from the larger set, [tex]\dbinom n2[/tex] subsets consisting of two members, and so on, which means the total number of possible subsets that can be chosen is
[tex]\displaystyle\binom n0+\binom n1+\cdots+\binom n{n-1}+\binom nn[/tex]
[tex]=\displaystyle\sum_{k=0}^n\binom nk[/tex]
Writing this as
[tex]=\displaystyle\sum_{k=0}^n\binom nk1^{n-k}1^k[/tex]
we can apply the binomial theorem, which states that this is equivalent to [tex](1+1)^n=2^n[/tex].
