Consider the following:

1. How many connected regions can $n$ hyperplanes form in $\mathbb R^d$?

2. What if the set of hyperplanes are homogeneous?

3. Given a set of $n$ pairs of hyperplanes, such that each pair is parallel, what is the maximum number of regions that can be formed?

I saw here,here and here that the answer to (1) is $$f(d,n)=\sum_{i=0}^d {n \choose i}$$

However I find it non-trivial to generalize the proof of $\mathbb R^2$ that was provided to $\mathbb R^d$ (without using "lower"/"upper" descriptions). Is there any "nice" way to show it recursively?

Q3 is what I'm really after.

Any ideas?