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 $k$.
Categories: math.NT