How did Abraham De Moivre contribute to combinatorial counting problems?

Abraham De Moivre made significant contributions to combinatorics, particularly in the field of enumeration problems. He is credited with identifying that the number of distinct non-negative integer solutions to the equation $x_1+x_2+\dots +x_n=m$ (where $m\ge n$) can be expressed using binomial coefficients.1

Beyond his work on specific equations, De Moivre advanced the mathematical tools necessary for combinatorial analysis. He originated Stirling’s formula, which provides an approximation for the factorial of large numbers ($n!$), a concept fundamental to calculating permutations and combinations. In 1733, he further utilized this formula to derive the normal frequency curve as an approximation of the binomial law, bridging the gap between combinatorial counting and probability theory. His work in The Doctrine of Chances also helped formalize the study of independent events, which remains a cornerstone of modern combinatorial probability.2