How many Hamilton circuits exist in 𝐾_10?

Question

i dont know how to show how many circuits are within K sub 10 and i need help with it. Im using the equation

n(n-1)/2 but i dont know if this is the right equation

Answer 1

Just to check, K_10 usually means the complete graph on 10 vertices---10 vertices where every pair of distinct vertices has an edge between them. Is that what you mean here? If not, disregard the rest of this answer.

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Sorry, n*(n-1)/2 isn't the right formula to count the Hamilton circuits of K_n.

For a Hamilton circuit, you need to visit every vertex once. To simplify things, let's pick a particular vertex and think of it as the start vertex for all of our circuits. I call it v1. To build a Hamilton circuit, you need to choose the next vertex to visit. We have 9 choices for that. Then we have to choose a third vertex, different from the start and different from the previous choice, but with no other restrictions since we're on K_10. How many options are there for that? How many choices for the fourth vertex? ... How many ways are there to build an ordered list of the 9 vertices after v1?

Now at the end, there's a slight wrinkle--we've double counted the circuits, since there are two different directions you can start listing the circuit starting from v1. So you will have to divide the answer you find above by 2.