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 Asia were visited by at least one of Dheeraj, Samantha and Nitesh?

Correct Answer: 3

To find the total number of countries in Asia visited by at least one of Dheeraj, Samantha, or Nitesh, we can analyze the data provided in the bar chart and the given facts step-by-step.

Step 1: Extract data from the chart for Asia
From the "Number of Countries visited" chart, we look at the blue solid bars representing the "Asia" region:
1. Dheeraj visited 3 countries in Asia.
2. Samantha has no bar for Asia, which means she visited 0 countries in Asia.
3. Nitesh visited 1 country in Asia.

Let us denote the set of countries in Asia visited by Dheeraj as D_A and by Nitesh as N_A. Since Samantha visited 0 countries in Asia, her set S_A is empty. Therefore, any country visited in Asia by at least one of them must belong to the union of Dheeraj's and Nitesh's visited Asian countries:
$$|D_A\cup N_A|$$

Step 2: Determine the intersection of countries visited in Asia
According to Fact 3: "China (in Asia) is the only country that was visited by both Dheeraj and Nitesh, but not by Samantha."
Since Samantha did not visit any countries in Asia, any country in Asia visited by both Dheeraj and Nitesh is automatically not visited by Samantha. Thus, the intersection of the sets of Asian countries visited by Dheeraj and Nitesh is exactly this single country (China).
Therefore, the number of common countries in Asia visited by both is:
$$|D_A\cap N_A|=1$$

Step 3: Calculate the union of visited Asian countries
Using the principle of inclusion-exclusion, the number of countries in Asia visited by at least one of Dheeraj, Samantha, and Nitesh is:
$$|D_A\cup N_A|=\left|D_A\right|+\left|N_A\right|-|D_A\cap N_A|$$

Substituting the values extracted from the chart and Fact 3:
$$|D_A\cup N_A|=3+1-1=3$$

Thus, there are exactly 3 countries in Asia visited by at least one of Dheeraj, Samantha, and Nitesh.