TITLE: What are the possible numbers of regions that 4 planes can divide space?
QUESTION [8 upvotes]: What are the possible numbers of regions that 4 planes can create? 
We know that the minimum number is 5 and the maximum number is 15.
(http://mathworld.wolfram.com/SpaceDivisionbyPlanes.html)
Is it possible to make a generalization based on the ways the planes 
could intersect?

REPLY [2 votes]: Computing the maximum number of pieces into which $r$ hyperplanes disconnect the space $\mathbb{R}^n$ is an old little problem, easily solved by induction, and the answer is
$${r \choose 0}+{r \choose 1}+\dots+{r \choose n},$$ and it is achieved if they are in generic position. So in dimension $n$ the number of pieces as $r$ increases is asymptotically $r^n/n!$. Note also that for $r\leq n$ it's $2^r$ and for $r=n+1$ it's $2^r-1$ (so in particular in dimension $3$ the number of pieces with $0,1,2,3,4$ generic planes is resp. $1,2,4,8$ and then $15$ (not $16$ as one would guess, so this may be used as an example of fallacious argument based on analogy).