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A quinpartite graph is a graph whose vertices can be partitioned into five
groups such that no two vertices in same group are connected via some edge.
The maximum number of edges in a quinpartite graph with 10 vertices,
where cardinalities of those five sets are given as {2,3,2,1,2}, is:
(A) 16
(B) 20
(C) 26
(D) 39
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There are Five Groups and there is no edge between member of same group ===> it is 5-pertite graph

Maximum no.of edges possible when it is complete 5-pertite graph.

let name the as A,B,C,D,E, and their cardinalities are k,l,m,n,p respectively.

the members of A should form a edge with all members of other groups.

No.of Edges = k * ( l+m+n+p)

the members of B should form a edge with all members of other groups (for avoiding duplicates eliminate A group )

No.of Edges = l * ( m+n+p )

the members of C should form a edge with all members of other groups (for avoiding duplicates eliminate A and B group )

No.of Edges = m * ( n+p )

the members of D should form a edge with all members of other groups (for avoiding duplicates eliminate A,B,C group )

No.of Edges = n * ( p)

the members of E should form a edge with all members of other groups (for avoiding duplicates eliminate A,B,C,and D group )

No.of Edges = p * ( 0 )

Total edges = ( k * ( l+m+n+p) ) + (  l * ( m+n+p ) )  + (  m * ( n+p ) )  + ( n * ( p) ) .

Fix first the cardinalities of groups, then evaluate, For your question, answer is 39

by Veteran (61.8k points)
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nicely explained Thanks :)
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You can refer http://faculty.kutztown.edu/rieksts/225-F-08/graphs/tripartite.html to know about a complete quintpartite graph :)

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