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Number of k -combinations for all k

The number of k -combinations for all k is the number of subsets of a set of n elements. There are several ways to see that this number is 2. In terms of combinations, ∑ 0 ≤ k ≤ n (n k) = 2 n {\textstyle \sum _{0\leq {k}\leq {n}}{\binom {n}{k}}=2^{n}}, which is the sum of the n th row (counting from 0) of the binomial coefficients in Pascal's triangle. These combinations (subsets) are enumerated by the 1 digits of the set of base 2 numbers counting from 0 to 2 − 1, where each digit position is an item from the set of n.

Given 3 cards numbered 1 to 3, there are 8 distinct combinations (subsets), including the empty set:

| { { }; { 1 }; { 2 }; { 1, 2 }; { 3 }; { 1, 3 }; { 2, 3 }; { 1, 2, 3 } } | = 2 3 = 8 {\displaystyle |\{\{\};\{1\};\{2\};\{1,2\};\{3\};\{1,3\};\{2,3\};\{1,2,3\}\}|=2^{3}=8}

Representing these subsets (in the same order) as base 2 numerals: