Abstract
With the rapid development of frontier neural network, artificial intelligence and big data technology, the related theory and application research of combinatorial mathematics are more and more extensive. The application of binomial coefficient in combinatorial mathematics is an effective way to solve the research of information coding and quantum computation. Binomial coefficients and its distributions are the core topic in probability statistics, and there are many theories and applications related to them.In this paper, the quantification of binomial coefficients and the characteristic of reflection rotation transformation are studied using three-dimensional diagrams. Using variant construction, the combinatorial clustering properties are investigated applying binomial formulas and sample distributions and their combinatorial patterns are illustrated. It is proved that the basic binomial coefficient formula and its extended model have obvious properties of reflection and rotation invariance.
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