Closed expressions are obtained for sums of products of Bernoulli numbers of the form \[ \sum {{2n} \choose {2j_1,\dots,2j_N}} B_{2j_1}\cdots B_{2j_N}, \] where the factor in parentheses is a multinomial coefficient and the summation is extended over all nonnegative integers \(j_1,\dots,j_N\) whose sum is \(n\). Corresponding results are derived for Bernoulli polynomials, and for Euler numbers and polynomials. Sums of this type were previously studied by Euler for \(n=2\), and by others for \(n=3\) and 4.