Although there are records from prehistory involving problems on the subject, the term "combinatorics" in the most modern sense of mathematics was first used in the paper. Combinatorial Art Dissertation (Dissertation on Combinational Arts), written by Gottfried Wilhelm Leibniz and published in 1666 in the city of Leipzig, Germany.
A curious fact is that this work was an extended version of Leibniz's first doctoral dissertation, written before the author had studied mathematics thoroughly. In 1690, the pamphlet was reprinted without Leibniz's consent, which led him to express his annoyance about its circulation, considering it immature at the time. Even so, it was an original work that gave it high visibility among the scholars of the time.
In English, the term "combinatory" was only to appear in 1818 in a work by Peter Nicholson (OED2) entitled Essays on the Combinatorial Analysis (Assay on combinatorial analysis).