  This paper studies the product $\bar{G}_n$ of the binomial coefficients in
the n-th row of Pascal's triangle, which equals the reciprocal of the product
of all the reduced and unreduced Farey fractions of order n. It studies its
size as a real number, measured by its logarithm $log(\bar{G}_n)$, and its
prime factorization, measured by the order of divisibility by a fixed prime p,
each viewed as a function of n. It derives three formulas for its prime power
divisibility, $ord_p(\bar{G}_n)$, two of which relate it to base p radix
expansions of n, and which display different facets of its behavior. These
formulas are used to determine the maximal growth rate of each
$ord_p(\bar{G}_n)$ and structure of the fluctuations of these functions. It
also defines analogous functions for all integer bases $b$ replacing prime
bases. A final topic relates the factorizations of $\bar{G}_n$ to
Chebyshev-type prime-counting estimates and the prime number theorem.
