Discrete Mathematics with Graph Theory (3rd Edition) 190

Discrete Mathematics with Graph Theory (3rd Edition) 190 -...

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Unformatted text preview: 188 Solutions to Exercises 18. By the Principle of Inclusion-Exclusion, the number of elements in at least one of the given sets is IAI U A2 U A3 U A4 U A51 = 5(23) - m (10) + m (4) - m (1) + 0 = 115 - 100 + 40 - 5 = 50. Thus, 52 - 50 = 2 elements belong to none. 19. (a) [BB] A line segment can be formed between any two points and if two pairs of points are different so are the lines they determine. The number of different line segments is C20) = 45. (b) For any choice of three points, a triangle is determined. Also, if two triples are different, so are the triangles they determine. So the number of triangles is C30) = 120. (c) A triangle with A as a vertex can be formed for any two points different from A, so the number is m =36. 20. (a) [BB] A triangle is determined by three of the 12 vertices. The answer is C3 2 ) = 220. (b) It is impossible for three sides of the polygon to form a triangle. There are 12 ways in which a triangle can be formed involving exactly two (necessarily adjacent) sides of the polygon (because...
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