A school awards 77 medals in three sports i.e. 48 in football, 25 in tennis and 25 in cricket. If 7 students got medals in all the three sports, then the number of students who received medals in exactly 2 sports is
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Solution
Let A, B and C be the sets of students who won medals in football, tennis and cricket respectively.
Then, n(A)=48,n(B)=25,n(C)=25,n(A∩B∩C)=7
and n(A∪B∪C)=77
We know that, n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C)
Let a,b and c be the number of students who won medals in football and tennis both, tennis and cricket both, football and cricket both respectively and d be the number of students who won medals in all three games.
Then, n(A∩B)+n(B∩C)+n(C∩A)=28
⇒(a+d)+(b+d)+(c+d)=28
⇒(a+b+c)+3d=28
⇒(a+b+c)+(3×7)=28{∵d=7}
⇒(a+b+c)=7
Hence, 7 students receive medals in exactly 2 sports.