"A 6-tuple contains 6 elements, not 20."
Well, yeah.
What I meant:
in the sixtuple (1,2,3,4,5,6) the triples
(1,2,3)(1,2,4)(1,2,5)(1,2,6)(1,3,4)(1,3,5)(1,3,6)(1,4,5)(1,4,6)(1,5,6)
(2,3,4)(2,3,5)(2,3,6)(2,4,5)(2,4,6)(2,5,6)(3,4,5)(3,4,6)(3,5,6)(4,5,6)
so 20 triples is the maximum that can be fit...