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In a group of 400 boys, each boy plays atleast one of the games cricket, hockey and football. 185 play…

Let \(x\) boys play all three games. Using the Venn diagram totals: \(\text{Only C} = 185 - (40 + 20 + x) = 125 - x\); \(\text{Only H} = 165 - (20 + 10 + x) = 135 - x\);…

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In a group of 400 boys, each boy plays atleast one of the games cricket, hockey and football. 185 play cricket, 165 play hockey and 160 play football. 40 boys play only cricket and football. 20 boys play only hockey and cricket. 10 boys play only hockey and football. How many boys play all the three games ?

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Correct answer A

Let xx boys play all three games. Using the Venn diagram totals: Only C=185−(40+20+x)=125−x\text{Only C} = 185 - (40 + 20 + x) = 125 - x; Only H=165−(20+10+x)=135−x\text{Only H} = 165 - (20 + 10 + x) = 135 - x; Only F=160−(40+10+x)=110−x\text{Only F} = 160 - (40 + 10 + x) = 110 - x. Summing all 7 disjoint regions: (125−x)+(135−x)+(110−x)+40+20+10+x=400  ⟹  440−2x=400  ⟹  x=20(125-x) + (135-x) + (110-x) + 40 + 20 + 10 + x = 400 \implies 440 - 2x = 400 \implies x = 20.

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  • RPSC Rajasthan State and Subordinate Services (RAS/RTS)

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