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CAT 2024 Slot 1 — DILR Question 14

3 SetsEasy
Passage / Data

Answer the following questions based on the information given below.

The chart below provides complete information about the number of countries visited by Dheeraj, Samantha and Nitesh, in Asia, Europe and the rest of the world (ROW).

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The following additional facts are known about the countries visited by them.
1. 32 countries were visited by at least one of them.
2. USA (in ROW) is the only country that was visited by all three of them.
3. China (in Asia) is the only country that was visited by both Dheeraj and Nitesh, but not by Samantha.
4. France (in Europe) is the only country outside Asia, which was visited by both Dheeraj and Samantha, but not by Nitesh.
5. Half of the countries visited by both Samantha and Nitesh are in Europe.

How many countries in Asia were visited by at least one of Dheeraj, Samantha and Nitesh?

Answer & solution

Answer: 3

Solution

Hard

First read the chart, then use the five facts with inclusion–exclusion to fix every overlap. The same set-up answers all four questions of this set.

0 6 12 Dheeraj Samantha Nitesh Asia Europe ROW

Shared by all four questions. Totals: Dheeraj 3+7+1=113+7+1=11, Samantha 0+9+4=130+9+4=13, Nitesh 1+6+12=191+6+12=19; sum =43=43. With triple t=1t=1 (USA), inclusion–exclusion gives 32=43−(a+b+c)−2t32=43-(a+b+c)-2t, so a+b+c=9a+b+c=9 where a=a= D&S-only, b=b= D&N-only, c=c= S&N-only. Fact 3 ⇒ b=1b=1 (China); Fact 4 + Samantha having no Asian country ⇒ a=1a=1 (France); hence c=7c=7 and ∣S∩N∣=8|S\cap N|=8, of which half (=4=4) are in Europe (Fact 5).

1

Count distinct Asian countries:

Asia: D=3, S=0, N=1(from chart)only overlap: China (D∩N)(Fact 3)⇒ distinct=3+1−1=3\begin{aligned} &\text{Asia: } D=3,\ S=0,\ N=1 \quad\text{(from chart)}\\ &\text{only overlap: China } (D\cap N) \quad\text{(Fact 3)}\\ &\Rightarrow\ \text{distinct}=3+1-1=3 \end{aligned}
3 countries\mathbf{3}\ \text{countries}
CAT 2024 Slot 1 DILR Q14: How many countries in Asia were visited by at least one of Dheeraj, Samantha and Nitesh? — Solution | TheCATExam