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Hint: In these types of questions remember to use the basic formula of the sets for example $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$ and solve the given question.
Complete step-by-step answer:
According to the given information
Let set A be the medals awarded in football n (A) = 38
Set B be the medals awarded in basketball n (B) = 15
Set C be the medals awarded in cricket n (C) = 20
And the total number of medals be U = 58 or the total medals awarded to men $n(A \cup B \cup C) = 58$
Now let the 3 men got the medals in all the sports $n(A \cap B \cap C) = 3$
Using the formula of $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$ ----- (equation 1)
Putting the values in the equation 1
$ \Rightarrow $$n(A \cap B) + n(B \cap C) + n(A \cap C) = 38 - 58 + 15 + 20 + 3$
$ \Rightarrow $$n(A \cap B) + n(B \cap C) + n(A \cap C) = 18$ ---(Equation 2)
Only for $(A \cap B)$= $n(A \cap B) - n(A \cap B \cap C)$ -- (equation 3)
Only for$(B \cap C)$=$n(B \cap C) - n(A \cap B \cap C)$ ---(equation 4)
Only for$(A \cap C)$=$n(A \cap C) - n(A \cap B \cap C)$ ---(equation 5)
For medals received by exactly two of the three sports adding equation 3, 4 and 5
$ \Rightarrow $$n(A \cap B) + n(B \cap C) + n(A \cap C) - 3n(A \cap B \cap C)$ --(Equation 6)
Putting the values in equation 6
$ \Rightarrow $18 – 3(3) = 9
Therefore medals received by exactly two of the three sports are 9 medals.
Note: In these types of questions first use the given information to find the given values then use the basic formulas of sets like $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$ and then use this formula to find the values of unknown factors like $n(A \cap B) + n(B \cap C) + n(A \cap C)$and then make the equation for each case of sports and then apply the addition operation in all the three equations the coming result is the answer of the question
Complete step-by-step answer:
According to the given information
Let set A be the medals awarded in football n (A) = 38
Set B be the medals awarded in basketball n (B) = 15
Set C be the medals awarded in cricket n (C) = 20
And the total number of medals be U = 58 or the total medals awarded to men $n(A \cup B \cup C) = 58$
Now let the 3 men got the medals in all the sports $n(A \cap B \cap C) = 3$
Using the formula of $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$ ----- (equation 1)
Putting the values in the equation 1
$ \Rightarrow $$n(A \cap B) + n(B \cap C) + n(A \cap C) = 38 - 58 + 15 + 20 + 3$
$ \Rightarrow $$n(A \cap B) + n(B \cap C) + n(A \cap C) = 18$ ---(Equation 2)
Only for $(A \cap B)$= $n(A \cap B) - n(A \cap B \cap C)$ -- (equation 3)
Only for$(B \cap C)$=$n(B \cap C) - n(A \cap B \cap C)$ ---(equation 4)
Only for$(A \cap C)$=$n(A \cap C) - n(A \cap B \cap C)$ ---(equation 5)
For medals received by exactly two of the three sports adding equation 3, 4 and 5
$ \Rightarrow $$n(A \cap B) + n(B \cap C) + n(A \cap C) - 3n(A \cap B \cap C)$ --(Equation 6)
Putting the values in equation 6
$ \Rightarrow $18 – 3(3) = 9
Therefore medals received by exactly two of the three sports are 9 medals.
Note: In these types of questions first use the given information to find the given values then use the basic formulas of sets like $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$ and then use this formula to find the values of unknown factors like $n(A \cap B) + n(B \cap C) + n(A \cap C)$and then make the equation for each case of sports and then apply the addition operation in all the three equations the coming result is the answer of the question
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