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  2. Combinatorial number system - Wikipedia

    en.wikipedia.org/wiki/Combinatorial_number_system

    (this associates distinct numbers to all finite sets of natural numbers); then comparison of k-combinations can be done by comparing the associated binary numbers. In the example C and C′ correspond to numbers 1001011001 2 = 601 10 and 1010001011 2 = 651 10, which again shows that C comes before C′.

  3. Combination - Wikipedia

    en.wikipedia.org/wiki/Combination

    These combinations (subsets) are enumerated by the 1 digits of the set of base 2 numbers counting from 0 to 2 n − 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 :

  4. Contract bridge probabilities - Wikipedia

    en.wikipedia.org/wiki/Contract_bridge_probabilities

    Hence, the number of suit permutations of the 4-4-3-2 pattern is twelve. Or, stated differently, in total there are twelve ways a 4-4-3-2 pattern can be mapped onto the four suits. Or, stated differently, in total there are twelve ways a 4-4-3-2 pattern can be mapped onto the four suits.

  5. Pascal's triangle - Wikipedia

    en.wikipedia.org/wiki/Pascal's_triangle

    For example, the number of 2-dimensional elements in a 2-dimensional cube (a square) is one, the number of 1-dimensional elements (sides, or lines) is 4, and the number of 0-dimensional elements (points, or vertices) is 4. This matches the 2nd row of the table (1, 4, 4).

  6. Stars and bars (combinatorics) - Wikipedia

    en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)

    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 coefficient ( n − 1 k − 1 ) = ( 10 − 1 4 − 1 ) = ( 9 3 ) = 84 , {\displaystyle {\binom {n-1}{k-1}}={\binom {10-1}{4-1}}={\binom {9}{3}}=84,}

  7. Permutation - Wikipedia

    en.wikipedia.org/wiki/Permutation

    This would have been the first attempt on record to solve a difficult problem in permutations and combinations. [4] Al-Khalil (717–786), an Arab mathematician and cryptographer, wrote the Book of Cryptographic Messages. It contains the first use of permutations and combinations, to list all possible Arabic words with and without vowels. [5]

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  9. Twelvefold way - Wikipedia

    en.wikipedia.org/wiki/Twelvefold_way

    Two lists count as the same if it is possible to both reorder and relabel them as above and produce the same result. For example, (3, 5, 3) and (2, 9, 9) count as the same because they can be reordered as (3, 3, 5) and (9, 9, 2) and then relabeling both produces the same list (1, 1, 2).