When you have three collinear points there is one gradient. I'm not sure what your question is specifically but when points are collinear they have the same gradient.
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I think you mean: Are any three points contained in exactly one plane? only if they're not collinear... I think
3 non-collinear points define one plane.
The definition of a non-collinear line is that this is a line on which points do not lie on one line. The opposite of this is a collinear point. Collinear points refer to three points that do fall on a straight line.
Only one if they are non-collinear. An infinite number if they are collinear.
If three points all lie on the same line, then the points are said to be "collinear". This is also true if the slope from each point to the next is the same.