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Infinitely many, though in a plane, only one straightline.

However, an arbitrarily large number of curved linescould pass through two points.

[As an aside there some non-Euclidian geometries in which the above do not apply, but I'm sure they lie outside your scope of inquiry.]

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Q: How many lines can go through two points?
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How many lines can you draw through two points?

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How many lines can be drawn through two fixed points?

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