The expected number of aggregates in a random distribution of n points

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Abstract
When n points are distributed at random (equally likely anywhere) over a large area or volume, they may form small clusters or aggregates. In this paper a k-aggregate is defined more precisely as k points which can be covered by a small figure of given size, shape and orientation, but whose position is not otherwise specified. Exact formulae for the expected number of such k-aggregates are given in some cases, while a good approximate formula (accurate in most practical cases) and limits are stated, covering all cases.