We work on $\mathbb{R}^n$ for a natural integer $n \geqslant 3$. Let $m \in \mathbb{N}^*$.
Show that the set
$$\left\{ (i_1, i_2, \ldots, i_n) \in \mathbb{N}^n \mid i_1 + i_2 + \cdots + i_n = m \right\}$$
has cardinality $\dbinom{n+m-1}{m}$. Deduce the dimension of $\mathcal{P}_m$.