Bijective proof

Let the sequence $\left\{ r_{n}\right\}$ be defined as follows:

Prove that the map $n \mapsto r_{n}$ is a bijection of the set $\mathbb {N}$ onto the set $\mathbb {Q}_{+}=\mathbb {Q} \cap (0,+\infty )$.

Let $\mathscr {F}(\mathbb {N})=\left\{ A \subset \mathbb {N} \mid \operatorname {card}(A)<+\infty \right\}$ be the system of all finite subsets of the set $\mathbb {N}$. Prove that

The number of zig-zag paths between the street corners $(0,0)$ and $(r,s)$ is

The number of those zig-zag paths between the street corners $(0,0)$ and $(r,s)$ the area under which is $\alpha$ equals $c_{n,r,\alpha }$ (notation of problem 60.5).

The number of inversions generated by a zig-zag path equals the area under the path. (Thus problem 60.10 determines the number of certain paths, or letter sequences, having a given number $\alpha$ of inversions.)

A multiset is an unordered collection of elements in which each element may occur any number of times. Two multisets are equal if each element occurs in them the same number of times. For example, the multiset $\left\{ 1,2,1,3\right\}$ equals the multiset $\left\{ 3,1,1,2\right\}$ and does not equal the multiset $\left\{ 1,2,2,3\right\}$.

Prove that the number of ways to form a $k$-element multiset from $n$ pairwise distinct elements, using each of them any number of times, equals $C_{n+k-1}^{k}$.