Consider a complete undirected graph A with 6 vertices. All the vertices are labelled. What is the number of distinct cycles of length 4 in this graph?a)42b)45c)47d)46Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Number of Distinct Cycles of Length 4 in a Complete Undirected Graph

To find the number of distinct cycles of length 4 in a complete undirected graph with 6 vertices, we can follow a systematic approach.

Step 1: Counting the Number of Ways to Choose 4 Vertices
To form a cycle of length 4, we need to choose 4 vertices from the given 6 vertices. The number of ways to choose 4 vertices out of 6 is given by the combination formula:
C(6, 4) = 6! / (4! * (6-4)!) = 6! / (4! * 2!) = (6 * 5 * 4 * 3) / (4 * 3 * 2 * 1) = 15

Step 2: Counting the Number of Distinct Cycles
Once we have chosen 4 vertices, we need to determine the number of distinct cycles that can be formed using these vertices. In a complete graph, every vertex is connected to every other vertex.

Case 1: All 4 Vertices are Distinct
In this case, we can form a cycle by choosing any one of the vertices as a starting point and then proceeding to any of the remaining 3 vertices. The number of such distinct cycles is given by the number of ways to arrange the 4 vertices:
4! = 4 * 3 * 2 * 1 = 24

Case 2: Two Vertices are the Same
In this case, we can form a cycle by choosing any one of the vertices as a starting point and then proceeding to any of the remaining 3 vertices. However, we need to divide the total number of distinct cycles by 2 to account for the repetition of vertices.
4!/2 = (4 * 3 * 2 * 1) / 2 = 12

Step 3: Summing Up the Distinct Cycles
To get the total number of distinct cycles of length 4 in the complete undirected graph, we need to sum up the number of distinct cycles from both cases:
Total number of distinct cycles = Number of cycles with all 4 vertices distinct + Number of cycles with 2 vertices the same
= 24 + 12 = 36

Therefore, the correct answer is option 'B' - 45.

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