Question

A school awarded 30 medals in tennis, 14 in carrom and 25 in badminton. If these medals were bagged by a total of 50 students and only 5 students got medals in all the three sports, then the number of medals received by students in exactly two of the three games is

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Solution

The correct option is **C** 9

Let T,C,B denote the sets of students who bagged medals in tennis, carrom and badminton respectively.

⇒n(T)=30;n(C)=14;n(B)=25

Also, n(T∪C∪B)=50;n(T∩C∩B)=5

Now, we know the formula:

n(T∪C∪B)=n(T)+n(C)+n(B)

−n(T∩C)−n(C∩B)−n(B∩T)

+n(C∩B∩T)

To find: Number of medals received by students in exactly 2 games.

n(T∩C)+n(T∩B)+n(B∩C)=x(let)

Inserting the values in the formula:

50=30+14+25−x+5

⇒x=30+14+25+5−50=24

Number of medals received by students in exactly 2 games=n(T∩C)+n(T∩B)+n(B∩C)−3n(T∩B∩C)=24−3(5)=9

Hence, 9 students bagged exactly two out of three medals.

Let T,C,B denote the sets of students who bagged medals in tennis, carrom and badminton respectively.

⇒n(T)=30;n(C)=14;n(B)=25

Also, n(T∪C∪B)=50;n(T∩C∩B)=5

Now, we know the formula:

n(T∪C∪B)=n(T)+n(C)+n(B)

−n(T∩C)−n(C∩B)−n(B∩T)

+n(C∩B∩T)

To find: Number of medals received by students in exactly 2 games.

n(T∩C)+n(T∩B)+n(B∩C)=x(let)

Inserting the values in the formula:

50=30+14+25−x+5

⇒x=30+14+25+5−50=24

Number of medals received by students in exactly 2 games=n(T∩C)+n(T∩B)+n(B∩C)−3n(T∩B∩C)=24−3(5)=9

Hence, 9 students bagged exactly two out of three medals.

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