This last fact can be seen directly, for a subset of a set of size n can be identified by a binary string of length nin the following way. We consider the set in question in a specific order{a1, a2,…, an}say,and then a binary string of length n specifies a subset by saying that each instance of 1 in the string indicates the presence of the corresponding ai in the subset in question. For example, ifn=4,the strings 0111 and 0000 stand respectively for{a2, a3, a4}, and for the empty set. Since there are two choices for each entry in the binary string, there are 2n such strings in all and therefore 2nsubsets within a set of size n.
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