In a class of 200 students 70 play cricket, 60 play hockey and 80 play football. 30 play cricket and football, 30 play hockey and football, 40 play cricket and hockey. If 130 people play at least one game, find the number of people who play all the three games.


This is reply by AI

Let I denote people who play one game only, II denote people who play 2 games only, III denote people who play all 3 games only

I + II + III + none = 200
none should be 70 as 130 play atleast one sports
I + II + III = 130
Also remember , the total of all the population in the cricket, hockey and football (The 3 venn diagrams without subtracting the duplicates) would be 210. This would be equal to,

I + 2II + 3III = 210 - why? check our sets article to understand
I + II + III = 130
II + 2III = 80
Now we know II = (30 - III) + (30 -III) + (40-III)
II = 100 - 3III
II = 80 - 2III
80 - 2III = 100 - 3III

so, III = 20

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