  We consider the problem of characterizing solutions in $(x, y)$ to the
equation ${x \choose y}={{x-a} \choose {y+b}}$ in terms of $a$ and $b$. We
obtain one simple result which allows the determination of a ratio in terms of
$a$ and $b$ which the ratio $\frac{x}{y}$ must approximate. We then add to the
understanding of the infinite family of repeated coefficients discovered by D.
Singmaster, by using fundamental results from Diophantine geometry to prove
that in the case $a \neq b$, solutions to ${x \choose y}={{x-a} \choose {y+b}}$
are finite. Finally, we make some observations about the potential utility of
equations of the form ${x \choose y}={{x-a} \choose {y+b}}$ in proving
Singmaster's conjecture, which is the main unsolved problem in the area of
repeated binomial coefficient study. We remark that this approach to the
conjecture is markedly different from previous approaches, which have only
established logarithmic bounds on a function which counts the number of
representations of $t$ as a binomial coefficient.
