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It can be used to solve a variety of counting problems, such as how many ways there are to put n indistinguishable balls into k distinguishable bins. [4] The solution to this particular problem is given by the binomial coefficient ( n + k − 1 k − 1 ) {\displaystyle {\tbinom {n+k-1}{k-1}}} , which is the number of subsets of size k − 1 ...
It is the unique solution to a two-person bargaining problem that satisfies the axioms of scale invariance, symmetry, efficiency, and independence of irrelevant alternatives. According to Paul Walker, [ 3 ] Nash's bargaining solution was shown by John Harsanyi to be the same as Zeuthen 's solution [ 4 ] of the bargaining problem.
Solution of a travelling salesman problem: the black line shows the shortest possible loop that connects every red dot. In the theory of computational complexity, the travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the ...
A minimum spanning tree of a weighted planar graph.Finding a minimum spanning tree is a common problem involving combinatorial optimization. Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, [1] where the set of feasible solutions is discrete or can be reduced to a discrete set.
A solution to Kirkman's schoolgirl problem with vertices denoting girls and colours denoting days of the week [1] Kirkman's schoolgirl problem is a problem in combinatorics proposed by Thomas Penyngton Kirkman in 1850 as Query VI in The Lady's and Gentleman's Diary (pg.48). The problem states:
Frobenius coin problem with 2-pence and 5-pence coins visualised as graphs: Sloping lines denote graphs of 2x+5y=n where n is the total in pence, and x and y are the non-negative number of 2p and 5p coins, respectively. A point on a line gives a combination of 2p and 5p for its given total (green).
Statutory Consolidation: a business combination that creates a new company in which none of the previous companies survive. Stock Acquisition: a business combination in which the purchasing company acquires the majority, more than 50%, of the Common stock of the acquired company and both companies survive. Variable interest entity
Combinatorial explosion is sometimes used to justify the intractability of certain problems. [ 1 ] [ 2 ] Examples of such problems include certain mathematical functions , the analysis of some puzzles and games, and some pathological examples which can be modelled as the Ackermann function .