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