Q: How many non-collinear points are there in one plane?

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no

Only one plane can pass through 3 non-collinear points.

exactly one and only one.

If you're asking a question, then the answer is 'no'. If you're making a statement, then the statement is false. I can always lay a single plane down on any three points you choose. If your points are in the same straight line, then there an infinite number of other planes that your points all lie in. If they're not all in the same straight line, then there's only one plane.

The set of all points in the plane equidistant from one point in the plane is named a parabola.

Related questions

just one

no

3 non-collinear points define one plane.

Yes. You require three non-collinear points to uniquely define a plane!

Only one plane can pass through 3 non-collinear points.

Yes a plane can always be drawn three any three points, whether they are linear or not.

One.exactly one

exactly one and only one.

Through any three noncollinear points, there is exactly one plane.

If you're asking a question, then the answer is 'no'. If you're making a statement, then the statement is false. I can always lay a single plane down on any three points you choose. If your points are in the same straight line, then there an infinite number of other planes that your points all lie in. If they're not all in the same straight line, then there's only one plane.

The answer depends on the number of point. One point - as the question states - cannot be non-collinear. Any two points are always collinear. But three or more points will define a plane. If four points are non-coplanar, they will define four planes (as in a tetrahedron).

A minimum of three points are required to define a plne (if they are not collinear). And in projective geometry you can have a plane with only 3 points. Boring, but true. In normal circumstances, a plane will have infinitely many points. Not only that, there are infinitely many in the tiniest portion of the plane.