  Let $G_n= \prod_{k=0}^n \binom{n}{k},$ the product of the elements of the
$n$-th row of Pascal's triangle. This paper studies the partial factorizations
of $G_n$ given by the product $G(n,x)$ of all prime factors $p$ of $G_n$ having
$p \le x$, counted with multiplicity. It shows $\log G(n, \alpha n) \sim
f_G(\alpha)n^2$ as $n \to \infty$ for a limit function $f_{G}(\alpha)$ defined
for $0 \le \alpha \le 1$. The main results are deduced from study of functions
$A(n, x), B(n,x),$ that encode statistics of the base $p$ radix expansions of
the integer $n$ (and smaller integers), where the base $p$ ranges over primes
$p \le x$. Asymptotics of $A(n,x)$ and $B(n,x)$ are derived using the prime
number theorem with remainder term or conditionally on the Riemann hypothesis.
