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In 1947 Fine obtained an expression for the number of binomial coefficientson row n of Pascal's triangle that are nonzero modulo p. In this paper we useKummer's theorem to generalize Fine's theorem to prime powers, expressing thenumber of nonzero binomial coefficients modulo p^alpha as a sum over certaininteger partitions. For fixed alpha, this expression can be rewritten to showexplicit dependence on the number of occurrences of each subword in the base-prepresentation of n.

The number of nonzero binomial coefficients modulo p^alpha