Question
Sat July 21, 2012 By: R.k. Kumar

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 three men got medals in all the three sports, how many received medals in exactly two of the three sports?

Expert Reply
Sat July 21, 2012
n(FUBUC) = n(F) + n(B) + n(C) - n(BnC) - n(FnB) - n(FnC) + n(FnBnC)
 
58 = 38 + 15 + 20 - [n(BnC) + n(FnB) + n(FnC)] + 3
 
18 = [n(BnC) + n(FnB) + n(FnC)]
 
Number of men who received medals in exactly two of the three sports
 =n(FnBnC') + n(FnB'nC) + n(F'nBnC)
 =n(BnC) + n(FnB) + n(FnC) - 3n(FnBnC)
 =9   {Since n(FnBnC') = n(FnB) - n(FnBnC)}
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