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 the ROW were visited by both Nitesh and Samantha?

The correct option is 4.

From the bar chart titled "Number of Countries visited", we can extract the number of countries visited by Dheeraj, Samantha, and Nitesh in each of the three regions: Asia, Europe, and Rest of the World (ROW).
• 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 us define the variables for the Venn diagram representing the countries visited by Dheeraj (D), Samantha (S), and Nitesh (N) in each region R (where R can be Asia, Europe, or ROW):
• Let d_R, s_R, and n_R represent the number of countries visited exclusively by D, S, and N, respectively, in region R.
• Let ds_R represent the number of countries visited by D and S but not N in region R.
• Let dn_R represent the number of countries visited by D and N but not S in region R.
• Let sn_R represent the number of countries visited by S and N but not D in region R.
• Let dsn_R represent the number of countries visited by all three in region R.

The sum of the total visits across all travelers is:
$$\text{Total Visits}=11+13+19=43$$

According to Fact 1, the total number of unique countries visited by at least one person is:
$$\text{Unique Countries}=32$$

Using Venn diagram relationships, we know:
$$\text{Total Visits}=N_1+2N_2+3N_3$$
$$\text{Unique Countries}=N_1+N_2+N_3$$
where $N_1$ is the number of countries visited by exactly one person, $N_2$ is the number of countries visited by exactly two people, and $N_3$ is the number of countries visited by all three.

Subtracting the two equations gives:
$$\text{Total Visits}-\text{Unique Countries}=N_2+2N_3$$
Substituting the values:
$$43-32=N_2+2N_3\Rightarrow N_2+2N_3=11$$

Fact 2 states that the USA (in ROW) is the only country visited by all three of them. This means:
$$N_3=1$$
Thus, we have ${\mathrm{dsn}}_{\text{ROW}}=1$ and ${\mathrm{dsn}}_{\text{Asia}}={\mathrm{dsn}}_{\text{Europe}}=0$.
Substituting $N_3=1$ into our equation:
$$N_2+2\left(1\right)=11\Rightarrow N_2=9$$

The total number of countries visited by exactly two people, $N_2$, is the sum:
$$N_2=\mathrm{dn}+\mathrm{ds}+\mathrm{sn}=9$$

Let's find the values of $\mathrm{dn}$ and $\mathrm{ds}$ using the remaining facts:
• According to Fact 3, China (in Asia) is the only country visited by D and N but not S. Thus, $\mathrm{dn}=1$ (with ${\mathrm{dn}}_{\text{Asia}}=1$).
• According to Fact 4, France (in Europe) is the only country outside Asia visited by D and S but not N. Since Samantha visited 0 countries in Asia, there are no countries in Asia visited by D and S. Thus, $\mathrm{ds}=1$ (with ${\mathrm{ds}}_{\text{Europe}}=1$).
Substituting these values:
$$1+1+\mathrm{sn}=9\Rightarrow \mathrm{sn}=7$$
Since Samantha visited 0 countries in Asia, ${\mathrm{sn}}_{\text{Asia}}=0$. This means:
$${\mathrm{sn}}_{\text{Europe}}+{\mathrm{sn}}_{\text{ROW}}=7$$

According to Fact 5, half of the countries visited by both Samantha and Nitesh are in Europe.
The total number of countries visited by both Samantha and Nitesh is:
$$\text{Total visited by S and N}=\mathrm{sn}+\text{dsn}=7+1=8$$
Thus, the number of countries visited by both in Europe is:
$$\frac{1}{2}\times 8=4$$
Since ${\mathrm{dsn}}_{\text{Europe}}=0$, all 4 of these countries must belong to ${\mathrm{sn}}_{\text{Europe}}$:
$${\mathrm{sn}}_{\text{Europe}}=4$$

Substituting this back into the equation:
$$4+{\mathrm{sn}}_{\text{ROW}}=7\Rightarrow {\mathrm{sn}}_{\text{ROW}}=3$$

The question asks for the total number of countries in the ROW visited by both Nitesh and Samantha. This includes the countries visited by both S and N but not D in ROW (${\mathrm{sn}}_{\text{ROW}}$), as well as the country visited by all three of them in ROW (${\mathrm{dsn}}_{\text{ROW}}$, which is the USA):
$$\text{Total ROW countries visited by both S and N}={\mathrm{sn}}_{\text{ROW}}+{\mathrm{dsn}}_{\text{ROW}}$$
$$\text{Total ROW countries visited by both S and N}=3+1=4$$
Thus, the number of countries in the ROW visited by both Nitesh and Samantha is 4.