How many countries in Europe were visited by exactly one of Dheeraj, Samantha and Nitesh?

	Asia	Europe	Rest of World	Total
Dheeraj	3	7	1	11
Samantha	0	9	4	13
Nitesh	1	6	12	19
Total	4	22	17	43

This table shows an overlap of countries (i.e. countries which is visited by all three or any two of them as well.

Now according to statement 1,

There were a total of 32 countries.

So using the basic venn diagram equations we get,

I + II + III = 32...(1)

I + 2 II + 3 III=43...(2)

Where:

• I denotes countries visited by only one of them
• II denotes countries visited by only 2 of them
• III denotes countries visited by all 3 of them.

Using statement 2:

As USA is the only country that was visited by all three of them.

Therefore, III = 1

Keeping the same in the equation we get:

I + II = 31

And, I + 2 II = 40

Solving both equations we will get:

II = 9

and I = 22

So we are looking for 22 countries visited by only one person and 9 countries visited by only 2 people.

We can shift to a more specific table to clarify the data.

Combination	Asia	EU	ROW	Total
Dheeraj
Samantha
Nitesh
Dheeraj and Nitesh	1(China)
Dheeraj and Samantha		1(France)
Samantha and Nitesh
All Three	0	0	1(USA)
Total				32

Using the information given in the set, we can fill this much of the data directly.

Combination	Asia	EU	ROW	Total
Dheeraj
Samantha
Nitesh
Dheeraj and Nitesh	1(China)	0	0	1
Dheeraj and Samantha	0	1(France)	0	1
Samantha and Nitesh
All Three	0	0	1(USA)
Total				32

Now here we know (using statement 3) that both dheeraj and nitesh has visited only 1 country i.e., China

Therefore, Europe and ROW will be 0.

Similarly we can say for dheeraj and samantha (using statement 4).

Combination	Asia	EU	ROW	Total
Dheeraj
Samantha
Nitesh
Dheeraj and Nitesh	1(China)	0	0	1
Dheeraj and Samantha	0	1(France)	0	1
Samantha and Nitesh				7
All Three	0	0	1(USA)
Total				32

Now we already know that II (countries visited by only 2) = 9 countries

And we know the total number of visits by Dheeraj and Nitesh both, and Dheeraj and Samantha both, which is 1 each.

So total number of countries visited by both samantha and nitesh will be:

9-1-1 = 7

Combination	Asia	EU	ROW	Total
Dheeraj
Samantha
Nitesh
Dheeraj and Nitesh	1(China)	0	0	1
Dheeraj and Samantha	0	1(France)	0	1
Samantha and Nitesh		4		7
All Three	0	0	1(USA)
Total				32

Now using statement 5:

Countries visited by both samantha and nitesh is 7 + 1 (USA) = 8

Now half of it is in Europe.

i.e., 8/2 = 4.

Combination	Asia	EU	ROW	Total
Dheeraj			0
Samantha	0
Nitesh
Dheeraj and Nitesh	1(China)	0	0	1
Dheeraj and Samantha	0	1(France)	0	1
Samantha and Nitesh	0	4	3	7
All Three	0	0	1(USA)
Total				32

Now we know that dheeraj visited only one country in ROW

therefore, every other value in that column will be 0.

Similarly Samantha has not visited any country in Asia.

Therefore, every value with Samantha present in Asia will be 0.

Therefore, for samantha and nitesh in ROW will be 7-4 = 3

Combination	Asia	EU	ROW	Total
Dheeraj	2	6	0	8
Samantha	0			4
Nitesh	0			10
Dheeraj and Nitesh	1(China)	0	0	1
Dheeraj and Samantha	0	1(France)	0	1
Samantha and Nitesh	0	4	3	7
All Three	0	0	1(USA)	1
Total	3	17	12	32

Reference table:

	ASIA	EUROPE	ROW
Dheeraj	3	7	1
Samantha	0	9	4
Nitesh	1	6	12

Now we already know Dheeraj has visited 3 places in Asia and one is China (visited by Dheeraj and nitesh together). Therefore, only left i.e., 3 - 1 = 2 will be visited by dheeraj alone.

Similarly we know Nitesh had visited only one country in asia which is china, therefore he will not visit any asian country alone.

Combination	Asia	EU	ROW	Total
Dheeraj	2	6	0	8
Samantha	0	4	0	4
Nitesh	0			10
Dheeraj and Nitesh	1(China)	0	0	1
Dheeraj and Samantha	0	1(France)	0	1
Samantha and Nitesh	0	4	3	7
All Three	0	0	1(USA)	1
Total	3	17	12	32

In Europe, we know Samantha has visited 9 countries out of which 1 is with dheeraj and 4 with Samantha. Therefore countries visited by samantha alone in europe will be 9 - 1 - 4 = 4

After this, we know EU row total is 17 so countries visited by nitesh alone in europe is 2. Which helps us find out that the countries visited by him alone in ROW is 8.

Combination	Asia	EU	ROW	Total
Dheeraj	2	6	0	8
Samantha	0	4	0	4
Nitesh	0	2	8	10
Dheeraj and Nitesh	1(China)	0	0	1
Dheeraj and Samantha	0	1(France)	0	1
Samantha and Nitesh	0	4	3	7
All Three	0	0	1(USA)	1
Total	3	17	12	32

Countries in Europe visited by exactly one of Nitesh, Samantha, and Dheeraj is 6 +4 + 2 = 12