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GATE | GATE CS 2012 | Question 36

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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









Answer: (B)


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) 


(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. 


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Last Updated : 28 Jun, 2021
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