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Each such subset forms a valid 2-simplex, and all are non-empty and distinct. Thus, the number of distinct 2-simplices (triangles) is $35$.
Question: A science policy advisor is organizing a roundtable discussion with 8 scientists from 3 different disciplines: 3 physicists, 3 chemists, and 2 biologists. If scientists of the same discipline are indistinguishable in seating labeling, how many distinct seating arrangements are possible around a circular table?
Now divide by 8 for rotational symmetry in circular arrangements:
Thus, there are $70$ distinct seating arrangements.
Question: An industrial designer offers custom phone cases with 5 color palettes and 4 pattern options. A customer picks exactly 3 different colors and 1 pattern, where the order of colors does not matter. How many unique case designs can be made?
Solution: First, choose 3 different colors from 5, where order does not matter:
Therefore, there are $40$ unique phone case designs possible.
Thus, there are 126 distinct ways to label the vertices.
The distances form an arithmetic sequence:
The average distance over five days is given by: