Math Is Fun Forum

Hi;

There are other ways to attack this problem but from a computational standpoint this is my favorite.

Four points are coplanar iff the determinant of the following matrix is 0.

http://mathworld.wolfram.com/Coplanar.html

Expanding the determinant and setting it to 0 we get:

Dividing by 2400

This is a linear Diophantine equation. The number of solutions to them can be solved by the use of generating functions.

The number of solutions of

with the constraints 

is equal to the coefficient of x^0 ( the constant term ) in the expansion of

This product is nothing more than algebra and can be done by hand methods but it is best done by computer. If you need to I will show you how to take it over to Alpha. This is the power of generating functions, a hard problem has been reduced to a computational one. The expansion of the above product yields:

We see that the constant term is 344 and so the answer is  344 / 8^4 = 43 / 512.

Now I am not saying this is the simplest solution to this problem but these methods have the virtue of being able to solve a great many combinatorics problems with little or no change.