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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 factorizationsof $G_n$ given by the product $G(n,x)$ of all prime factors $p$ of $G_n$ having$p \le x$, counted with multiplicity. It shows $\log G(n, \alpha n) \simf_G(\alpha)n^2$ as $n \to \infty$ for a limit function $f_{G}(\alpha)$ definedfor $0 \le \alpha \le 1$. The main results are deduced from study of functions$A(n, x), B(n,x),$ that encode statistics of the base $p$ radix expansions ofthe 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 primenumber theorem with remainder term or conditionally on the Riemann hypothesis.

Partial Factorizations of Products of Binomial Coefficients