There is not really a concept of a “true beginning” in this type of mathematics. The beginning is defined to be wherever you start.
Where does a circle start and end? Every point on a circle is both the beginning and the end of that circle.
The math of permutations and combinatorials are closely related to the geometry of circles and because of that they take on a lot of similar properties. Another way to phase this question is “If I have a set of beads that are all the same size but different colors, and I want to chain multiple beads of different colors together using specific limited pallets of colors to make bracelets, what is the bracelet with the smallest size where after one rotation around the bracelet I will have seen every possible combination of the set of colors used in the pallet for that bracelet.” If that sounds confusing, it’s because it is.
But you don’t start from the beginning each time.
More of a philosophical question I suppose, not mathematical
There is not really a concept of a “true beginning” in this type of mathematics. The beginning is defined to be wherever you start.
Where does a circle start and end? Every point on a circle is both the beginning and the end of that circle. The math of permutations and combinatorials are closely related to the geometry of circles and because of that they take on a lot of similar properties. Another way to phase this question is “If I have a set of beads that are all the same size but different colors, and I want to chain multiple beads of different colors together using specific limited pallets of colors to make bracelets, what is the bracelet with the smallest size where after one rotation around the bracelet I will have seen every possible combination of the set of colors used in the pallet for that bracelet.” If that sounds confusing, it’s because it is.