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We give a heuristic argument supporting conjectures of Bhargava on theasymptotics of the number of $S_n$-number fields having bounded discriminant.We then make our arguments rigorous in the case $n=3$ giving a new elementaryproof of the Davenport-Heilbronn theorem. Our basic method is to count elementsof small height in $S_n$-fields while carefully keeping track of the index ofthe monogenic ring that they generate.

Heuristics for the asymptotics of the number of $S_n$-number fields