Infinitely many. There are infinitely many points in the plane and although any pair of points define a line, no matter how many lines you are given, it is always possible to find a point that is not on any of them - that is, a point that is not collinear.
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Any three non-collinear points will define a single plane. A plane is composed of an infinite number of distinct lines.
Just one plane.
For instance a plane.
A plane
it means a series of points that are not on the same line in a plane.