How many planes are determined by "n" points, no four of which are coplaner?
To determine the number of planes formed by "n" points, no four of which are coplanar, you can use the formula for the number of combinations of "n" objects taken four at a time. Since four points are required to uniquely define a plane, we need to find the number of combinations of "n" points taken four at a time.
The formula to calculate the number of combinations of "n" objects taken "r" at a time is:
C(n, r) = n! / (r! * (n - r)!)
In this case, we want to find C(n, 4), which represents the number of combinations of "n" points taken four at a time.
So, the number of planes determined by "n" points, no four of which are coplanar, is given by:
C(n, 4) = n! / (4! * (n - 4)!)
Let's work through an example to illustrate this:
Suppose we have 6 points. Using the formula, we can calculate the number of planes formed by these points:
C(6, 4) = 6! / (4! * (6 - 4)!)
= 6! / (4! * 2!)
= (6 * 5 * 4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * (2 * 1))
= 15
Therefore, for 6 points, there are 15 planes determined, no four of which are coplanar.