Multiple tetrahedra?

Probability Level 4

{x+y+z=1x+y+z=6x−y−z=1x−y−z=6y−z−x=1y−z−x=6z−y−x=1z−y−x=6 \begin{cases} x+y+z = 1\\ x+y+z = 6\\ x-y-z = 1\\ x-y-z = 6\\ y-z-x = 1\\ y -z-x = 6\\ z-y-x = 1\\ z-y-x = 6\\ \end{cases}

The above equations define planes in 3 space.

How many ways can you choose four of them so that the enclosed region defines a regular tetrahedron?

Image credit: Wikiwand
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Geoff Pilling by Geoff Pilling
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