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For any pair of positive integers n and k, the number of k-tuples of positive integers whose sum is n is equal to the number of (k − 1)-element subsets of a set with n − 1 elements. For example, if n = 10 and k = 4 , the theorem gives the number of solutions to x 1 + x 2 + x 3 + x 4 = 10 (with x 1 , x 2 , x 3 , x 4 > 0 ) as the binomial ...
The number associated in the combinatorial number system of degree k to a k-combination C is the number of k-combinations strictly less than C in the given ordering. This number can be computed from C = {c k, ..., c 2, c 1} with c k > ... > c 2 > c 1 as follows.
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 n . 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 ...
A Langford pairing for n = 4.. In combinatorial mathematics, a Langford pairing, also called a Langford sequence, is a permutation of the sequence of 2n numbers 1, 1, 2, 2, ..., n, n in which the two 1s are one unit apart, the two 2s are two units apart, and more generally the two copies of each number k are k units apart.
Combinations and permutations in the mathematical sense are described in several articles. Described together, in-depth: Twelvefold way; Explained separately in a more accessible way: Combination; Permutation; For meanings outside of mathematics, please see both words’ disambiguation pages: Combination (disambiguation) Permutation ...
The total number of combinations giving that result is, as stated above, the first number multiplied by the second. The expression is therefore ( 6 n ) ( 43 6 − n ) ( 49 6 ) {\displaystyle {6 \choose n}{43 \choose 6-n} \over {49 \choose 6}} .
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