into how many parts will 5 random planes divide space?

Question

this is polya's problem and i have to do it for geometry and it is the freakin hardest thing ever! someone please help me!

the answer is not 32 its 26, but i have absolutly no idea how to get it...

Answer 1

n planes divide space into
(n^3 + 5n + 6)/6 regions.
for n=5, the number of regions is 26.

To see why, consider smaller dimensions first.

First consider a line. n points on a line will divide the line into n+1 regions. So the number of regions for n = 1,2,3,4,... is:
2, 3, 4, 5, ...

Next, consider a plane. Into how many regions will n lines divide the plane? Well, lets say you have n lines already and you add an (n+1)st line. That line can cross the n other lines. When n lines are crossed n+1 new regions are formed. Therefore, the sequence above (2,3,4,5,6,...) for the line problem form the common differences in the sequence for the plane problem
i.e.,
2, 4, 7, 11, 16, ...
....2, 3, 4, 5, ...

For space, the same logic applies. For each new (the n+1st) plane, the number of new regions is increased by the number of regions a plane can be divided by n lines. The common differences here are then the numbers in the sequence above.
i.e.,
2, 4, 8, 15, 26, 42, ...
...2, 4, 7, 11, 16, ...
......2, 3, 4, 5, ...

This is by no means a formal proof but that's the gist of it.

Answer 2

I got 22

Answer 3

it is NOT 32~~~!!!
THE answer is 26, but the problem is how to find it :/

Answer 4

I think it is 5! ways (5 factorial)

5! = 5 * 4 * 3 * 2 * 1
5! = 120 ways

Answer 5

wow! this is a very gross project!!! i luv katie and sarah