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Example 34 - A college awarded 38 medals in football, 15 basketball

Example 34 - Chapter 1 Class 11 Sets - Part 2
Example 34 - Chapter 1 Class 11 Sets - Part 3 Example 34 - Chapter 1 Class 11 Sets - Part 4

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Example 34 A college awarded 38 medals in football, 15 in basketball and 20 in cricket. If these medals went to a total of 58 men and only 3 men got medals in all the three sports, how many received medals in exactly two of the three sports ? Let F , B, C denote the set of people who received medals in football , basketball & cricket respectively Number of medals won in football = n(F) = 38, Number of medals won in basketball = n (B) = 15, Number of medals won in cricket = n (C) = 20 Number of medals won in either football,basketball or cricket = n (F ∪ B ∪ C) = 58 Number of medals won in all football,basketball and cricket = n (F ∩ B ∩ C) = 3 We know n (F ∪ B ∪ C ) = n(F) + n(B) + n(C) – n(F ∩ B) – n (F ∩ C ) – n (B ∩ C ) + n ( F ∩ B ∩ C ) 58 = 38 + 15 + 20 – n(F ∩ B) – n (F ∩ C ) – n (B ∩ C ) + 3 58 = 38 + 15 + 20 + 3 – n(F ∩ B) – n (F ∩ C ) – n (B ∩ C ) 58 = 76 – n(F ∩ B) – n (F ∩ C ) – n (B ∩ C ) n(F ∩ B) + n (F ∩ C ) + n (B ∩ C) = 76 – 58 n (F ∩ B ) + n (F ∩ C ) + n (B ∩ C ) = 18 We need to calculate how many received medals in exactly two of the three sports In Venn Diagram U denotes the universal set F denotes the set of medals won in football B denotes the set of medals won in basketball C denotes the set of medals won in cricket Also, In the Venn diagram, Let a denote the number of men who got medals in football & basketball but not cricket Let b denote the number of men who got medals in football & cricket but not basketball Let c denote the number of men who got medals in cricket & basketball but not football Let d denote the number of men who got medals in football & basketball & cricket Here d = n (F ∩ B ∩ C) ∴ d = 3 Number of people who got exactly two medals = a + b + c From (1) , n (F ∩ B) + n (F ∩ C) + n (B ∩ C) = 18 (a + d) + (b + d) + (c + d) = 18 a + b + c + d + d + d = 18 a + b + c + 3d = 18 a + b + c + 3 × 3 = 18 a + b + c + 9 = 18 a + b + c = 18 – 9 a + b + c = 9 ∴ 9 people received medals in exactly two of the three sports

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.