[2] The correct number 3264 was found in about 1859 by Ernest de Jonquières who did not publish because of Steiner's reputation, and by Chasles using his theory of characteristics,[3] and by Berner in 1865.
However these results, like many others in classical intersection theory, do not seem to have been given complete proofs until the work of Fulton and MacPherson in about 1978.
So to find the correct answer, one has to somehow eliminate the plane of spurious degenerate conics from this calculation.
The total transform of a degree 6 hypersurface is 6H, and Steiner calculated (6H)5 = 65P as H5=P (where P is the class of a point in the Chow ring).
Fulton & MacPherson gave a precise description of exactly what "general position" means (although their two propositions about this are not quite right, and are corrected in a note on page 29 of their paper).