Sets, Relations and FunctionsMCQPYQ Dec 23Question 1905 of 217
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In a survey of 100\displaystyle 100 boys it was found that 50\displaystyle 50 used white shirts, 40\displaystyle 40 red shirts and 30\displaystyle 30 blue shirts. 20\displaystyle 20 were habituated in using both white and red shirts. 15\displaystyle 15 were using both red and blue shirts and 10\displaystyle 10 were using blue and white shirts. Find the number of boys who are using all colours.

Options

A20
B25
C30
D35
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Correct Answer

Option b25

All Options:

  • A20
  • B25
  • C30
  • D35

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Detailed Solution & Explanation

Let W\displaystyle W be the set of boys using white shirts, R\displaystyle R be the set of boys using red shirts, and B\displaystyle B be the set of boys using blue shirts.
We are given:
- Total number of boys: n(WRB)=100\displaystyle n(W \cup R \cup B) = 100 (assuming all boys in the survey use at least one color)
- n(W)=50\displaystyle n(W) = 50
- n(R)=40\displaystyle n(R) = 40
- n(B)=30\displaystyle n(B) = 30
- n(WR)=20\displaystyle n(W \cap R) = 20
- n(RB)=15\displaystyle n(R \cap B) = 15
- n(WB)=10\displaystyle n(W \cap B) = 10
We need to find the number of boys who use all three colors, which is represented by the intersection n(WRB)\displaystyle n(W \cap R \cap B).
Using the Principle of Inclusion-Exclusion for three sets:
n(WRB)=n(W)+n(R)+n(B)n(WR)n(RB)n(WB)+n(WRB)n(W \cup R \cup B) = n(W) + n(R) + n(B) - n(W \cap R) - n(R \cap B) - n(W \cap B) + n(W \cap R \cap B)100=50+40+30201510+n(WRB)100 = 50 + 40 + 30 - 20 - 15 - 10 + n(W \cap R \cap B)100=12045+n(WRB)100 = 120 - 45 + n(W \cap R \cap B)100=75+n(WRB)100 = 75 + n(W \cap R \cap B)n(WRB)=10075=25n(W \cap R \cap B) = 100 - 75 = 25
Hence, **Option B** is the correct answer.

About This Chapter: Sets, Relations and Functions

Paper

Paper 3: Quantitative Aptitude

Weightage

3-5 Marks

Key Topics

Sets, Relations, Functions

This chapter covers Sets, Relations, Functions and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 3-5 Marks weightage. Focus on understanding core concepts rather than memorizing.

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