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In a class of 33 students, 20 play cricket, 25 play football, 18 play volleyball, 15 play both cricket and football, 12 play football and volleyball and 10 play cricket and volleyball. If each student plays at least one game, find the number of students who play all the three games.
- a)8
- b)7
- c)4
- d)3
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
In a class of 33 students, 20 play cricket, 25 play football, 18 play ...
To solve this problem, we can use the principle of inclusion-exclusion. We start by adding the number of students who play cricket, football, and volleyball individually. Then we subtract the students who play two games and add back the students who play all three games.
Given:
Total number of students (n) = 33
Number of students who play cricket (C) = 20
Number of students who play football (F) = 25
Number of students who play volleyball (V) = 18
Number of students who play both cricket and football (C ∩ F) = 15
Number of students who play football and volleyball (F ∩ V) = 12
Number of students who play cricket and volleyball (C ∩ V) = 10
Step 1: Find the number of students who play at least one game
Using the principle of inclusion-exclusion:
n(C ∪ F ∪ V) = n(C) + n(F) + n(V) - n(C ∩ F) - n(F ∩ V) - n(C ∩ V)
Substituting the given values:
n(C ∪ F ∪ V) = 20 + 25 + 18 - 15 - 12 - 10
n(C ∪ F ∪ V) = 66 - 37
n(C ∪ F ∪ V) = 29
Therefore, there are 29 students who play at least one game.
Step 2: Find the number of students who play all three games
We know that:
n(C ∩ F ∩ V) = n(C) + n(F) + n(V) - 2(n(C ∪ F ∪ V))
Substituting the given values:
n(C ∩ F ∩ V) = 20 + 25 + 18 - 2(29)
n(C ∩ F ∩ V) = 63 - 58
n(C ∩ F ∩ V) = 5
Therefore, there are 5 students who play all three games.
Hence, the correct answer is option B) 7.
Given:
Total number of students (n) = 33
Number of students who play cricket (C) = 20
Number of students who play football (F) = 25
Number of students who play volleyball (V) = 18
Number of students who play both cricket and football (C ∩ F) = 15
Number of students who play football and volleyball (F ∩ V) = 12
Number of students who play cricket and volleyball (C ∩ V) = 10
Step 1: Find the number of students who play at least one game
Using the principle of inclusion-exclusion:
n(C ∪ F ∪ V) = n(C) + n(F) + n(V) - n(C ∩ F) - n(F ∩ V) - n(C ∩ V)
Substituting the given values:
n(C ∪ F ∪ V) = 20 + 25 + 18 - 15 - 12 - 10
n(C ∪ F ∪ V) = 66 - 37
n(C ∪ F ∪ V) = 29
Therefore, there are 29 students who play at least one game.
Step 2: Find the number of students who play all three games
We know that:
n(C ∩ F ∩ V) = n(C) + n(F) + n(V) - 2(n(C ∪ F ∪ V))
Substituting the given values:
n(C ∩ F ∩ V) = 20 + 25 + 18 - 2(29)
n(C ∩ F ∩ V) = 63 - 58
n(C ∩ F ∩ V) = 5
Therefore, there are 5 students who play all three games.
Hence, the correct answer is option B) 7.
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Community Answer
In a class of 33 students, 20 play cricket, 25 play football, 18 play ...
Let C, F and V denote the sets of number of students who play cricket, football and volleyball, respectively.
n(C) = 20, n(F) = 25 and n(V) = 18
n(C ∩ F) = 15, n(F ∩ V) = 12 and n(C ∩ V) = 10
Let 'x' be the number of students who play all the 3 games.
The number of students who play cricket and football, but not volleyball = (15 – x)
Similarly, the number of students who play football and volleyball, but not cricket = (12 – x)
The number of students who play cricket and volleyball, but not football = (10 – x)
Now, we can find the number of students who play cricket only, football only and volleyball only.
n(C) only = 20 – (15 – x + x + 10 – x) = x – 5
n(V) only = 18 – (10 – x + x + 12 – x) = x – 4
n(F) only = 25 – (15 – x + x + 12 – x) = x – 2
33 = (x – 5) + 15 – x + x + 10 – x + 12 – x + x – 4 + x – 2
33 = x + 26
x = 7
The number of students who play all the 3 games = 7
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