  We study the minimum number of maximum matchings in a bipartite multigraph G
with parts $X$ and $Y$ under various conditions, refining the well-known lower
bound due to M. Hall. When $|X|=n$, every vertex in $X$ has degree at least
$k$, and every vertex in $X$ has at least $r$ distinct neighbors, the minimum
is $r!(k-r+1)$ when $n\ge r$ and is $[r+n(k-r)]\prod_{i=1}^{n-1}(r-i)$ when
$n<r$. When every vertex has at least two neighbors and $|Y|-|X|=t\ge 0$, the
minimum is $[(n-1)t+2+b](t+1)$, where $b=|E(G)|-2(n+t)$. We also determine the
minimum number of maximum matchings in several other situations. We provide a
variety of sharpness constructions.
