Let's solve problem 43, which asks:

"Чотири точки не лежать в одній площині. Скільки площин можна провести через трійку цих точок?"

Translation: "Four points do not lie in the same plane. How many planes can be passed through a triplet of these points?"

Solution:

1. Key Concept:

   • Any three non-collinear points define a unique plane.
   • Since the four points do not lie in the same plane, no four points can be co-planar.

2. Choosing Triplets:

   • From four points, we can select a group of three points in several ways.

3. Calculating the Number of Planes:

   • The number of ways to choose 3 points out of 4 is calculated by the combination formula $\binom{4}{3}$.

   $\binom{4}{3} = \frac{4!}{3!(4-3)!} = 4$

Thus, 4 different planes can be defined by choosing triplets of points from the four points given.

Would you like more details or have any questions?

Related Questions:

1. What if all four points did lie in the same plane—how many planes could you define?
2. How does the concept of collinearity affect the number of planes that can be defined by three points?
3. Can more than one plane contain three given points? Why or why not?
4. How many distinct lines can be drawn between four non-collinear points?
5. What changes in the number of planes if one of the four points lies on a plane formed by the other three?

Tip:

Understanding how planes and lines are defined by points is fundamental in geometry, especially when dealing with higher-dimensional spaces.