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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 3 sports, how many received medals in exactly two of the 3 sports?

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 3 sports, how many received medals in exactly two of the 3 sports? Correct Answer 9

Given:

A college awarded 38 medals in Football, 15 in Basketball and 20 in Cricket

Total medals won = 58 

3 men got medals in all the 3 sports

Formula Used:

n(AUBUC) = n(A) + n(B) + n(C) - n(AꓵB) - n(BꓵC) - n(AꓵC) + n(AꓵBꓵC)

Calculation:

Let the number of medal in football be n(F) = 38

Let the number of medal in basketball be n(B) = 15

Let the number of medal in cricket be n(C) = 20

n(FꓵBꓵC) = 3

n(FUBUC) = 58

According to the question,

n(FUBUC) = n(F) + n(B) + n(C) - n(FꓵB) - n(BꓵC) - n(FꓵC) + n(FꓵBꓵC)

⇒ 58 - 3 = 38 + 15 + 20 - n(FꓵB) - n(BꓵC) - n(FꓵC) 

⇒ n(FꓵB) + n(BꓵC) + n(FꓵC)  = 3 + 38 + 15 + 20 - 58 

⇒ n(FꓵB) + n(BꓵC) + n(FꓵC) = 18 

Now, for then number of who are getting medals of exactly in 2 sports out of 3 sports

⇒ only (FꓵB) + only (BꓵC) + only (FꓵC) = n(FꓵB) + n(BꓵC) + n(FꓵC) - 3 × n(FꓵBꓵC)

⇒ only (FꓵB) + only (BꓵC) + only (FꓵC) = 18 - 3 × 3 = 9 

∴ 9 men will get medals in exactly 2 sports out of 3 sports

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