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We give an exact coefficients formula of any infinite product of power serieswith constant term equal to $1$, by using structures from partitions ofintegers and permutation groups. This is an universal theorem for various ofBinomial-type theorems in many sense. In particular, we give the new formulasas the double counting of Bell polynomial, Binomial Theorem and MultinomialTheorem.

A Generalization of Bell Polynomials and Multinomial Expansions via  Permutations on Partitions, by Perturbation expansions of Functional  Determinants