Competition problem — Combinatorics

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Combinatorics Difficulty 5.0 AIME, harder Find the answer

The set of points $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ in $\mathbf{R}^{4}$ such that $x_{1} \geq x_{2} \geq x_{3} \geq x_{4}$ is a cone (or hypercone, if you insist). Into how many regions is this cone sliced by the hyperplanes $x_{i}-x_{j}=1$ for $1 \leq i<j \leq n$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

$C(4)=14$ .

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.