Question: Can we prove the correctness of the inequality $\sqrt{(x^2+y^2+z^2)}\leq a \sqrt{(x^2+y^2)}+b|z|$ where $x, y, z\in \mathbb{R}.$ What are such $a$ and $b?$

Answer: 
For every subset of a set $\mathcal{M}$ with $M$ elements except for $\emptyset$ and $\mathcal{M}$ one gets a partition with two nonempty sets, and there are $2^M-2$ such subsets. Since every (non-ordered) parttition is counted twice in this way You divide by 2 to obtain

$n=2^{M-1}-1$.
What Melech said is correct, but I think the heart of the matter is why there are $2^M$ subsets of a set of size M. That can be seen as follows. Firstly there are $M \choose K $ subsets of size K. So one needs to calculate $ \sum\_{K=0}^{K=M} {M\choose K}$. But, by the binomial theorem, this is just $ (1+1)^M = 2^M$ . Now insert what Melech said.