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](https://math.stackexchange.com/questions/409518/how-many-resulting-regions-if-we-partition-mathbbrm-with-n-hyperplanes/409642#409642),[here](https://math.stackexchange.com/questions/843834/trying-to-understand-formula-for-counting-regions-of-hyperplane-arrangements-in?rq=1) and [here](https://mathoverflow.net/questions/17202/sum-of-the-first-k-binomial-coefficients-for-fixed-n/17261#17261) 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?