Directions (Q.38-Q.41): 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).

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 Europe were visited only by Nitesh?

The correct answer is 2.

Step-by-step Explanation:

First, let's extract the total number of countries visited by Dheeraj, Samantha, and Nitesh in each region (Asia, Europe, and the Rest of the World / ROW) from the bar chart:
• Dheeraj: Asia = 3, Europe = 7, ROW = 1 (Total = 11)
• Samantha: Asia = 0, Europe = 9, ROW = 4 (Total = 13)
• Nitesh: Asia = 1, Europe = 6, ROW = 12 (Total = 19)

Let the number of countries visited in Europe be represented as:
• $d_E$: Visited only by Dheeraj
• $s_E$: Visited only by Samantha
• $n_E$: Visited only by Nitesh
• $d{s}_E$: Visited by Dheeraj and Samantha, but not Nitesh
• $d{n}_E$: Visited by Dheeraj and Nitesh, but not Samantha
• $s{n}_E$: Visited by Samantha and Nitesh, but not Dheeraj
• $ds{n}_E$: Visited by all three

Similarly, we can define variables for ROW (with subscript $R$) and Asia (with subscript $A$).

1. Analyzing Asia:
Since Samantha visited 0 countries in Asia, all countries visited in Asia were visited only by Dheeraj, only by Nitesh, or by both Dheeraj and Nitesh.
According to Fact 3, "China (in Asia) is the only country that was visited by both Dheeraj and Nitesh, but not by Samantha."
Thus, the country visited by both Dheeraj and Nitesh in Asia is exactly 1 (China).
Since Nitesh visited only 1 country in Asia, that country must be China.
Therefore, the number of countries visited only by Nitesh in Asia is 0.
Since Dheeraj visited 3 countries in Asia, and 1 of them is China, the other 2 countries in Asia were visited only by Dheeraj.
Total distinct countries visited in Asia = 3.

2. Analyzing ROW:
According to Fact 2, "USA (in ROW) is the only country that was visited by all three of them."
Thus, $ds{n}_R=1$ (USA).
According to Fact 4, "France (in Europe) is the only country outside Asia, which was visited by both Dheeraj and Samantha, but not by Nitesh."
This means that in ROW, there are no countries visited by both Dheeraj and Samantha but not Nitesh. Thus, $d{s}_R=0$.
Since Dheeraj visited only 1 country in ROW in total, and that country is the USA (visited by all three), Dheeraj visited no other countries in ROW.
This gives:
• $d_R=0$
• $d{n}_R=0$
For Samantha in ROW:
$s_R+s{n}_R+1=4\Rightarrow s_R+s{n}_R=3$
For Nitesh in ROW:
$n_R+s{n}_R+1=12\Rightarrow n_R+s{n}_R=11$
Thus, the total number of distinct countries visited in ROW is:
$\text{Distinct ROW}=s_R+n_R+s{n}_R+1=(3-s{n}_R)+(11-s{n}_R)+s{n}_R+1=15-s{n}_R$

3. Analyzing Europe:
Since USA is the only country visited by all three, we have $ds{n}_E=0$.
According to Fact 4, France is the only country outside Asia visited by both Dheeraj and Samantha but not Nitesh, so $d{s}_E=1$ (France).
According to Fact 3, China is the only country in the world visited by both Dheeraj and Nitesh but not Samantha.
Therefore, in Europe, there can be no such country. Thus, $d{n}_E=0$.
Using the total numbers for Europe:
• For Dheeraj: $d_E+d{s}_E+d{n}_E=7\Rightarrow d_E+1+0=7\Rightarrow d_E=6$
• For Samantha: $s_E+d{s}_E+s{n}_E=9\Rightarrow s_E+1+s{n}_E=9\Rightarrow s_E+s{n}_E=8$
• For Nitesh: $n_E+d{n}_E+s{n}_E=6\Rightarrow n_E+0+s{n}_E=6\Rightarrow n_E+s{n}_E=6$
Thus, the total number of distinct countries visited in Europe is:
$\text{Distinct Europe}=d_E+s_E+n_E+d{s}_E+d{n}_E+s{n}_E$
Substituting the values:
$\text{Distinct Europe}=6+(8-s{n}_E)+(6-s{n}_E)+1+0+s{n}_E=21-s{n}_E$

4. Using the Remaining Facts:
According to Fact 5, "Half of the countries visited by both Samantha and Nitesh are in Europe."
The countries visited by both Samantha and Nitesh are either in Europe ($s{n}_E$) or in ROW ($s{n}_R+1$, which includes USA).
Therefore:
$s{n}_E=\frac{1}{2}(s{n}_E+s{n}_R+1)\Rightarrow s{n}_E=s{n}_R+1\Rightarrow s{n}_R=s{n}_E-1$
According to Fact 1, there are 32 distinct countries in total:
$\text{Distinct Asia}+\text{Distinct Europe}+\text{Distinct ROW}=32$
$3+(21-s{n}_E)+(15-s{n}_R)=32$
$39-s{n}_E-(s{n}_E-1)=32$
$40-2s{n}_E=32$
$2s{n}_E=8\Rightarrow s{n}_E=4$

5. Finding the Answer:
We need to find the number of countries in Europe visited only by Nitesh ($n_E$):
$n_E=6-s{n}_E=6-4=2$
Thus, the number of countries in Europe visited only by Nitesh is 2.