The least significant bit in the binary numbers is on the left side!The (80)=1{\displaystyle {\tbinom {8}{0}}=1} 0-subsets, (81)=8{\displaystyle {\tbinom {8}{1}}=8} 1-subsets and (82)=28{\displaystyle {\tbinom {8}{2}}=28} 2-subsets of an 8-set,
the corresponding numbers (0, powers of two,  A018900)
and the coordinates in the simplices

The same for the (83)=56{\displaystyle {\tbinom {8}{3}}=56} 3-subsets:
Own work
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