Question

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

Answer 1

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