CAT 2024 Slot 1 DILR Set 2: Venn Diagrams (4 Questions) with Solutions

Samantha visited 0 Asian countries, so only Dheeraj and Nitesh matter for Asia.

Nitesh visited 1 Asian country. Clue 3 says China is visited by both Dheeraj and Nitesh but not Samantha, so Nitesh's only Asian country is China.

Dheeraj visited 3 Asian countries, and China is one of them. The other 2 are visited only by Dheeraj, because Nitesh has no other Asian country and Samantha has none.

Asian countries visited by at least one person = China + 2 only-Dheeraj countries = $3$.

Europe totals are Dheeraj 7, Samantha 9 and Nitesh 6.

Clue 2 says only USA (in ROW) is common to all three, so no European country is visited by all three.

Clue 3 says China (Asia) is the only Dheeraj-Nitesh-not-Samantha country, so Europe has none of these.

Clue 4 says France is the only non-Asian country visited by Dheeraj and Samantha but not Nitesh. So Europe has exactly 1 such country.

Dheeraj's 7 European countries are therefore France plus 6 visited only by Dheeraj.

Let $x$ be the number of European countries visited by both Samantha and Nitesh only. Then Samantha only = $9-1-x = 8-x$ and Nitesh only = $6-x$.

Europe union = $6+1+x+(8-x)+(6-x) = 21-x$.

Let $y$ be the number of ROW countries visited by both Samantha and Nitesh, excluding USA. In ROW, Dheeraj has only USA. The union is $1+(3-y)+y+(11-y)=15-y$.

Clue 1 gives $3+(21-x)+(15-y)=32$, so $x+y=7$.

Clue 5 says the Samantha-Nitesh overlap is $x+y+1$ (Europe $x$, ROW $y$ plus USA, Asia 0), and half of it is in Europe. So $x=\frac{x+y+1}{2}$, which gives $x=y+1$.

Solving gives $x=4$ and $y=3$.

Europe visited only by Nitesh = $6-4=2$.

Clue 2 puts USA (ROW) with all three. Dheeraj's ROW bar is 1, so USA is his only ROW country.

Clue 4 says France is the only non-Asian country visited by Dheeraj and Samantha but not Nitesh, and France is in Europe. Clue 3 says China is the only Dheeraj-Nitesh-not-Samantha country, and China is in Asia. So no European country is visited by Dheeraj and Nitesh without Samantha.

Let $x$ be the number of Europe-only Samantha-Nitesh countries and $y$ the number of ROW Samantha-Nitesh countries other than USA.

Europe: Dheeraj only = $7-1=6$, France = 1, Samantha only = $8-x$, Nitesh only = $6-x$. The union is $21-x$.

ROW: Samantha has $4-1-y=3-y$ countries only, Nitesh has $12-1-y=11-y$ only, and the union is $1+(3-y)+y+(11-y)=15-y$.

Asia has 3 countries, because Samantha has none and Dheeraj's 3 include China.

Clue 1 gives $3+(21-x)+(15-y)=32$, so $x+y=7$.

Samantha and Nitesh share $x+y+1$ countries (the extra 1 is USA). Clue 5 says half are in Europe, so $x=\frac{x+y+1}{2}$ and $x=y+1$. Hence $x=4$ and $y=3$.

ROW countries visited by both Nitesh and Samantha = $y+1$ (USA) $=4$.

Europe totals are Dheeraj 7, Samantha 9 and Nitesh 6.

Clue 2 says only USA (ROW) is common to all three, so Europe has no all-three country.

Clue 3 leaves no European country visited by Dheeraj and Nitesh without Samantha, because China (Asia) is the only such country.

Clue 4 says France is the only non-Asian Dheeraj-Samantha-not-Nitesh country, so Europe has 1 such country.

Dheeraj only = $7-1=6$.

Let $x$ be the number of Europe-only Samantha-Nitesh countries. Then Samantha only = $9-1-x=8-x$ and Nitesh only = $6-x$.

Let $y$ be the number of ROW Samantha-Nitesh countries other than USA. Clue 1 gives $3+(21-x)+(15-y)=32$, so $x+y=7$.

Clue 5 says the Samantha-Nitesh overlap is $x+y+1$ and half of it is in Europe. So $x=\frac{x+y+1}{2}$ and $x=y+1$, which gives $x=4$ and $y=3$.

Samantha only = $8-4=4$ and Nitesh only = $6-4=2$.

Europe countries visited by exactly one person = $6+4+2=12$.