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Consider a set, S, consisting of n elements.
Then the number of subsets of S containing r objects is equal to the number of ways of choosing r elements out of n = nCr.Therefore
nC0 + nC1 + nC2 + ... + nCn is the total number of subsets of S.
Now consider each subset of S. Each element of S can either be in the subset or not in the subset. So for each element there are two choices. Therefore, n elements give 2^n possible subsets.
Since these are two different approaches to the total number of subsets of S,
nC0 + nC1 + nC2 + ... + nCn = 2^n.
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