Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Let G be a complete undirected graph on 6 ver...
Start Learning for Free
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?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
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.
Attention Computer Science Engineering (CSE) Students!
To make sure you are not studying endlessly, EduRev has designed Computer Science Engineering (CSE) study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Computer Science Engineering (CSE).
|
Explore Courses for Computer Science Engineering (CSE) exam
|
|
Top Courses for Computer Science Engineering (CSE)View all
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?
Question Description
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? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about 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? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?.
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? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about 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? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?.
Solutions for 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? in English & in Hindi are available as part of our courses for Computer Science Engineering (CSE).
Download more important topics, notes, lectures and mock test series for Computer Science Engineering (CSE) Exam by signing up for free.
Here you can find the meaning of 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? defined & explained in the simplest way possible. Besides giving the explanation of
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?, a detailed solution for 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? has been provided alongside types of 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? theory, EduRev gives you an
ample number of questions to practice 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? tests, examples and also practice Computer Science Engineering (CSE) tests.
|
|
Explore Courses for Computer Science Engineering (CSE) exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup