If A, B, C are three arbitrary points, then the number of planes passing through these points is:

Question

Accepted Answer

Explanation:

1. **Why Option 1 is Correct:**
   - A plane in three-dimensional space is uniquely determined by three non-collinear points. When given three arbitrary points A, B, and C, if they are not collinear (i.e., they do not all lie on a single straight line), they define exactly one plane. This is because a plane is the two-dimensional surface that extends infinitely in all directions and is defined by the minimum number of points needed to establish a flat surface, which is three. Therefore, if A, B, and C are not collinear, there is exactly one plane that passes through all three points.

2. **Why the Other Options are Incorrect:**
   - **Option 2 (2 planes):** This option is incorrect because, as explained, three non-collinear points define exactly one plane. If the points were collinear, they would not define a plane at all, rather than defining multiple planes. Therefore, it is not possible for two distinct planes to pass through the same set of three non-collinear points.
   
   - **Option 3 (3 planes):** Similarly, this option is incorrect because three non-collinear points cannot define more than one plane. The concept of a plane is such that it is uniquely determined by three points, assuming they are not collinear. Thus, having three planes through the same three points is not possible.
   
   - **Option 4 (4 planes):** This option is also incorrect for the same reasons as above. The definition of a plane in geometry is such that three points (not lying on a single line) determine exactly one plane. Therefore, having four planes through the same three points is not feasible.

In conclusion, Option 1 is the correct answer because three non-collinear points uniquely determine one plane, and no more than one plane can pass through these points.