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Let G be a complete undirected graph on 6 vertices. If vertices of G are labeled, then the number of distinct cycles of length 4 in G is equal to
- a)15
- b)30
- c)45
- d)360
Correct answer is option 'C'. Can you explain this answer?
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Let G be a complete undirected graph on 6 vertices. If vertices of G a...
There can be total 6C4 ways to pick 4 vertices from 6. The value of 6C4 is 15. Note that the given graph is complete so any 4 vertices can form a cycle. There can be 6 different cycle with 4 vertices. For example, consider 4 vertices as a, b, c and d. The three distinct cycles are cycles should be like this (a, b, c, d,a) (a, b, d, c,a) (a, c, b, d,a) (a, c, d, b,a) (a, d, b, c,a) (a, d, c, b,a) and (a, b, c, d,a) and (a, d, c, b,a) (a, b, d, c,a) and (a, c, d, b,a) (a, c, b, d,a) and (a, d, b, c,a) are same cycles. So total number of distinct cycles is (15*3) = 45. **NOTE**: In original GATE question paper 45 was not an option. In place of 45, there was 90.
Most Upvoted Answer
Let G be a complete undirected graph on 6 vertices. If vertices of G a...
Explanation:
To solve this problem, we can use the concept of combinations and permutations.
Step 1: Counting the number of ways to select 4 vertices from 6
To form a cycle of length 4, we need to select 4 vertices from the 6 available vertices. The order in which we select these vertices does not matter.
The number of ways to select 4 vertices from 6 can be calculated using the combination formula:
nCr = n! / (r!(n-r)!)
In this case, n = 6 (number of vertices) and r = 4 (number of vertices to be selected).
6C4 = 6! / (4!(6-4)!) = 6! / (4!2!) = (6 * 5 * 4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * (2 * 1)) = 15
So, there are 15 ways to select 4 vertices from 6.
Step 2: Counting the number of ways to arrange the selected vertices
Once we have selected the 4 vertices, we need to arrange them in a cycle. The order in which we arrange these vertices matters.
The number of ways to arrange the 4 selected vertices can be calculated using the permutation formula:
nP4 = n! / (n-r)!
In this case, n = 4 (number of selected vertices) and r = 4 (number of vertices to be arranged).
4P4 = 4! / (4-4)! = 4! / 0! = 4! / 1 = 4 * 3 * 2 * 1 = 24
So, there are 24 ways to arrange the 4 selected vertices.
Step 3: Counting the number of distinct cycles
Finally, we need to count the number of distinct cycles that can be formed using the selected and arranged vertices.
Since the graph is complete, each cycle of length 4 can be formed in 2 distinct ways (clockwise and anticlockwise).
Therefore, the number of distinct cycles of length 4 in G is equal to:
Total number of ways to select and arrange = 15 * 24 = 360
Number of distinct cycles = Total number of ways to select and arrange / 2 = 360 / 2 = 180
Hence, the correct answer is option 'D', 360.
To solve this problem, we can use the concept of combinations and permutations.
Step 1: Counting the number of ways to select 4 vertices from 6
To form a cycle of length 4, we need to select 4 vertices from the 6 available vertices. The order in which we select these vertices does not matter.
The number of ways to select 4 vertices from 6 can be calculated using the combination formula:
nCr = n! / (r!(n-r)!)
In this case, n = 6 (number of vertices) and r = 4 (number of vertices to be selected).
6C4 = 6! / (4!(6-4)!) = 6! / (4!2!) = (6 * 5 * 4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * (2 * 1)) = 15
So, there are 15 ways to select 4 vertices from 6.
Step 2: Counting the number of ways to arrange the selected vertices
Once we have selected the 4 vertices, we need to arrange them in a cycle. The order in which we arrange these vertices matters.
The number of ways to arrange the 4 selected vertices can be calculated using the permutation formula:
nP4 = n! / (n-r)!
In this case, n = 4 (number of selected vertices) and r = 4 (number of vertices to be arranged).
4P4 = 4! / (4-4)! = 4! / 0! = 4! / 1 = 4 * 3 * 2 * 1 = 24
So, there are 24 ways to arrange the 4 selected vertices.
Step 3: Counting the number of distinct cycles
Finally, we need to count the number of distinct cycles that can be formed using the selected and arranged vertices.
Since the graph is complete, each cycle of length 4 can be formed in 2 distinct ways (clockwise and anticlockwise).
Therefore, the number of distinct cycles of length 4 in G is equal to:
Total number of ways to select and arrange = 15 * 24 = 360
Number of distinct cycles = Total number of ways to select and arrange / 2 = 360 / 2 = 180
Hence, the correct answer is option 'D', 360.
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Let G be a complete undirected graph on 6 vertices. If vertices of G are labeled, then the number of distinct cycles of length 4 in G is equal toa)15b)30c)45d)360Correct answer is option 'C'. Can you explain this answer?
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