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.