  In this note, we consider asymptotic products of binomial and multinomial
coefficients and determine their asymptotic constants and formulas. Among them,
special cases are the central binomial coefficients, the related Catalan
numbers, and binomial coefficients in a row of Pascal's triangle. For the
latter case, we show that it can also be derived from a limiting case of
products of binomial coefficients over the rows. The asymptotic constants are
expressed by known constants, for example, the Glaisher-Kinkelin constant. In
addition, the constants lie in certain intervals that we determine precisely.
Subsequently, we revisit a related result of Hirschhorn and clarify the given
numerical constant by showing the exact expression.
