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In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations).For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange.
The cake number is so-called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake. It is the 3D analogue of the lazy caterer's sequence. The values of C n for n = 0, 1, 2, ... are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, ... (sequence A000125 in the OEIS).
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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.
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An archetypal double counting proof is for the well known formula for the number () of k-combinations (i.e., subsets of size k) of an n-element set: = (+) ().Here a direct bijective proof is not possible: because the right-hand side of the identity is a fraction, there is no set obviously counted by it (it even takes some thought to see that the denominator always evenly divides the numerator).
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 coefficient
Perfect division – the number of pieces equals the number of agents: the cake should be partitioned into n pieces, and all agents agrees that all pieces have equal values. ε {\displaystyle \varepsilon } -near-exact division , for any constant ε > 0 {\displaystyle \varepsilon >0} , the agents may disagree on the pieces values, but the ...