Graphical approach may help

Algebra Level 3

Suppose there is a class with 30 students. Is it possible that 9 students have 3 friends each in the class, 11 have 4 friends each, and 10 have 5 friends each?

Yes No

Achal Jain by Achal Jain , 19, India

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1 solution

Kushal Bose
Aug 17, 2016

Consider the students as vertex of a graph and their friends as degree of that vertex.

  • 9 vertices has degree of each is 3.
  • 11 vertices has degree of each is 4
  • 10 vertices has degree of each is 5

So , total degree of all vertices is = ( 9 × 3 ) + ( 11 × 4 ) + ( 10 × 5 ) = 121 =(9 \times 3) + (11 \times 4) +(10 \times 5)=121

But in graph theory sum of degree of vertices is twice the number of edges i.e. even.But here it is odd

That's why not possible.

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