  We show that the space of linear series of certain multi-degree (including
the balanced ones) and rank $r$ on a general binary curve has the expected
dimension if nonempty. This generalizes Theorem 24 of Caporaso's paper about
binary curves from the case $r\leq 2$ to arbitrary rank, and shows that the
space of Osserman-limit linear series on a general binary curve has the
expected dimension, which was known for $r\leq 2$. In addition, we show that
this space of linear series is still of expected dimension after imposing
certain ramification conditions with respect to a sequence of increasing
effective divisors supported on two general points lying on different
components of the curve.
