Question

in a class of 60 students,23 play hockey,15 play basketball,20 play cricket and 7 play hockey and basketball,5 play cricket and basketball,4 play hockeyand cricket,15 do not play any of the three games. find the number of students who 1- play all the three games 2-play hockey but not cricket 3-play hockey and cricket both,but not basketball

Answer 1

Let B, C and H be the sets of students that plays Basketball, Cricket and Hockey

n(H) = 23, n(B) = 15, n(C) = 20

n(H intersection B) = 7, n(C intersection B) = 5, n(H intersection C) = 4

n(H union B union C) = 60 – 15 = 45

1. Hence, students that plays all games,

n(H intersection B intersection C) = n(H union B union C) – n(H) – n(B) – n(C) + n(H intersection B) + n(B intersection C) + n(C intersection H)

= 45 – 23 – 15 – 20 + 7 + 5 + 4 = 3

2. Who plays hockey not cricket

n(H) – n(H intersection C) = 23 – 4 = 19

3. Plays cricket and hockey not basketball

n(H intersection C) – n(H intersection C intersection B)

4 -3 = 1

Answer 2

Given :  In a class of 60 students of XI,

23 play hockey, 15 play basketball, 20 play cricket and 7 play hockey and basketball, 5 play cricket and basketball, 4 play hockey and cricket, 15 do not play any of the three games.  

To Find :  1. Students play hockey, basketball and cricket  together

2. Students play hockey but not cricket

3. Students play hockey and cricket but not basketball

4. Students play cricket but not hockey

5. Students play hockey and basketball but not cricket

Solution:

n ( Total) = 60

n(H) = 23

n (B) = 15

n(C) = 20

n ( H ∩ B) = 7

n ( H ∩ C) = 4

n ( C ∩ B) = 5

n ( H ∩ B ∩ C ) = ?

n (none) = 15

n ( Total) = n(H)  + n (B)  + n(C) - n ( H ∩ B) - n ( H ∩ C)  - n ( C ∩ B) + n ( H ∩ B ∩ C )  + n (none)

=> 60 = 23 + 15 + 20 - 7 - 4 - 5 +  n ( H ∩ B ∩ C ) + 15

=>  n ( H ∩ B ∩ C ) = 3

Students play hockey, basketball and cricket  together = n ( H ∩ B ∩ C ) = 3

Students play hockey but not cricket  

= n(H) -  n ( H ∩ C)

= 23 - 4

= 19

Students play hockey and cricket but not basketball  

= n ( H ∩ C)  -  n ( H ∩ B ∩ C )  

= 4 - 3

= 1

Students play cricket but not hockey  

= n(C) - n ( H ∩ C)

= 20 - 4

= 16

Students play hockey and basketball but not cricket

=  n ( H ∩ B)  -  n ( H ∩ B ∩ C )  

= 7 - 3

= 4

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