Final answer:
In a graph with 10 vertices, the number of Hamilton circuits is 9!, which is 362,880, and this count is the same irrespective of the starting vertex since each circuit can be rotated to start from any other vertex. option b is correct.
Step-by-step explanation:
The question involves finding the number of Hamilton circuits starting from each vertex in a graph with 10 vertices. A Hamilton circuit is a circuit that visits each vertex exactly once and returns to the starting vertex. To solve this, we can use the formula for permutations of (n-1) vertices for a Hamilton circuit, which is (n-1)! for a complete graph. Since the graph has 10 vertices, we start by finding the number of Hamilton circuits from one vertex, which is (10-1)!, or 9!.
Now, because the question asks for the number of Hamilton circuits starting from each vertex, we would multiply the result by the number of vertices. However, because Hamilton circuits are cycles, starting the cycle at a different vertex doesn't change the cycle, it's just rotated. Hence, the number of unique Hamilton circuits remains the same no matter which vertex we start from.
Therefore, the total number of Hamilton circuits starting from each vertex in a graph of 10 vertices is 9!, which is the number of Hamilton circuits from one vertex since each circuit can be rotated to start from any other vertex without creating a new unique circuit. Answer option b) 9, is the correct answer, implying 9! circuits which is 362,880 possible distinct Hamilton circuits.