  In this paper, we investigate the existence of Sierpi\'{n}ski numbers and
Riesel numbers as binomial coefficients. We show that for any odd positive
integer $r$, there exist infinitely many Sierpi\'{n}ski numbers and Riesel
numbers of the form $\binom{k}{r}$. Let $S(x)$ be the number of positive
integers $r$ satisfying $1\leq r\leq x$ for which $\binom{k}{r}$ is a
Sierpi\'{n}ski number for infinitely many $k$. We further show that the value
$S(x)/x$ gets arbitrarily close to 1 as $x$ tends to infinity. Generalizations
to base $a$-Sierpi\'{n}ski numbers and base $a$-Riesel numbers are also
considered. In particular, we prove that there exist infinitely many positive
integers $r$ such that $\binom{k}{r}$ is simultaneously a base
$a$-Sierpi\'{n}ski and base $a$-Riesel number for infinitely many $k$.
