No. The tiniest piece of a plane contains an infinite number of points. But if you
give us just three points, then we know exactly what plane you're talking about,
and it can't be any other plane.
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Exactly one.
Infinitely many planes contain any two given points- it takes three (non-collinear) points to determine a plane.
Only one plane can contain three specific points.
Exactly one.
Three non-co-linear points are sufficient to uniquely define a single plane.