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In a group of 265 persons, 200 like singing, 110 like dancing and 55 like painting, if 60 person like both singing and dancing, 30 like both singing and painting and 10 like all three activities then the number of person who like only dancing and painting is
- a)20
- b)10
- c)30
- d)40
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
In a group of 265 persons, 200 like singing, 110 like dancing and 55 l...
Solution: (1)
Let the number of people who like singing be n (S) = 200, who like dancing is n (D) = 110, number of people who like painting = n (P) = 55.
n (D ∪ P ∪ S) = 265
The number of people who like both singing and dancing = n (S ∩ D) = 60
The number of people who like both singing and painting = n (S ∩ P) = 30 and
The number of people who like all the 3 activities = n (D ∩ P ∩ S) = 10
n (D ∪ P ∪ S) = n (D) + n (P) + n (S) − n (D ∩ P) − n (P ∩ S) − n (S ∩ D) + n (D ∩ P ∩ S)
265 = 110 + 55 + 200 − n (D ∩ P) − 30 − 60 + 10
265 = 285 − n (D ∩ P)
n (D ∩ P) = 20
The number of persons who like dancing and painting = n (D ∩ P) − n (D ∩ P ∩ S)
= 20 − 10
= 10
This question is part of UPSC exam. View all Mathematics courses
Most Upvoted Answer
In a group of 265 persons, 200 like singing, 110 like dancing and 55 l...
Understanding the Problem
In a group of 265 persons, we need to analyze the preferences for three activities: singing, dancing, and painting. The data given is as follows:
Applying the Principle of Inclusion-Exclusion
To find the number of people who like only dancing and painting, we define:
- Let A = People who like singing
- Let B = People who like dancing
- Let C = People who like painting
We are looking for the number of people who like both dancing and painting but not singing. This can be calculated using the formula:
People who like both B and C = (People who like dancing) + (People who like painting) - (People who like both singing and dancing) - (People who like both singing and painting) + (People who like all three activities)
Using the data:
Now we can calculate:
Only those who like dancing and painting:
Formula:
\[ |B \cap C| = |B| + |C| - |A \cap B| - |A \cap C| + |A \cap B \cap C| \]
Substituting the values:
\[ |B \cap C| = 110 + 55 - 60 - 30 + 10 \]
Calculating:
\[ |B \cap C| = 110 + 55 - 60 - 30 + 10 = 85 \]
Now, to find those who like only dancing and painting:
\[ |B \cap C| - |A \cap B \cap C| = 85 - 10 = 75 \]
Since we want those who like only dancing and painting:
\[ |Dancing \cap Painting| - |A \cap B \cap C| = 75 - 10 = 20 \]
Thus, the number of people who like only dancing and painting is:
Final Answer
Therefore, the correct answer is option 'B'.
In a group of 265 persons, we need to analyze the preferences for three activities: singing, dancing, and painting. The data given is as follows:
- 200 like singing
- 110 like dancing
- 55 like painting
- 60 like both singing and dancing
- 30 like both singing and painting
- 10 like all three activities
Applying the Principle of Inclusion-Exclusion
To find the number of people who like only dancing and painting, we define:
- Let A = People who like singing
- Let B = People who like dancing
- Let C = People who like painting
We are looking for the number of people who like both dancing and painting but not singing. This can be calculated using the formula:
People who like both B and C = (People who like dancing) + (People who like painting) - (People who like both singing and dancing) - (People who like both singing and painting) + (People who like all three activities)
Using the data:
- Total who like dancing (B) = 110
- Total who like painting (C) = 55
- People who like both singing and dancing (A ∩ B) = 60
- People who like both singing and painting (A ∩ C) = 30
- People who like all three activities (A ∩ B ∩ C) = 10
Now we can calculate:
Only those who like dancing and painting:
Formula:
\[ |B \cap C| = |B| + |C| - |A \cap B| - |A \cap C| + |A \cap B \cap C| \]
Substituting the values:
\[ |B \cap C| = 110 + 55 - 60 - 30 + 10 \]
Calculating:
\[ |B \cap C| = 110 + 55 - 60 - 30 + 10 = 85 \]
Now, to find those who like only dancing and painting:
\[ |B \cap C| - |A \cap B \cap C| = 85 - 10 = 75 \]
Since we want those who like only dancing and painting:
\[ |Dancing \cap Painting| - |A \cap B \cap C| = 75 - 10 = 20 \]
Thus, the number of people who like only dancing and painting is:
Final Answer
- Option 'B' = 10
Therefore, the correct answer is option 'B'.
Community Answer
In a group of 265 persons, 200 like singing, 110 like dancing and 55 l...
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In a group of 265 persons, 200 like singing, 110 like dancing and 55 like painting, if 60 person like both singing and dancing, 30 like both singing and painting and 10 like all three activities then the number of person who like only dancing and painting isa)20b)10c)30d)40Correct answer is option 'B'. Can you explain this answer?
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