Question 147
Consider five-dimensional Cartesian space
$$
\mathbb{R}^{5}=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) \mid x_{i} \in \mathbb{R}\right\}
$$
and consider the hyperplanes with the following equations:
- $x_{i}=x_{j}$ for every $1 \leq i \lt j \leq 5$;
- $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=-1$;
- $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=0$;
- $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=1$.
Into how many regions do these hyperplanes divide $\mathbb{R}^{5}$?
$$
\mathbb{R}^{5}=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) \mid x_{i} \in \mathbb{R}\right\}
$$
and consider the hyperplanes with the following equations:
- $x_{i}=x_{j}$ for every $1 \leq i \lt j \leq 5$;
- $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=-1$;
- $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=0$;
- $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=1$.
Into how many regions do these hyperplanes divide $\mathbb{R}^{5}$?