There are 210 persons in a party, and all of them eat three different types of flavoured icecreams. 40 people eat only butterscotch, 30 people eat all three flavoured icecream,there are total 130 people who eat butterscotch and 100 people who eat vanila. 4 0 people eat butterscotch and vanila only, 10 people eat chocolate and vanila only.

What is number of people who eat only Chocolate?

The correct option is 50.

Step-by-Step Explanation:

Let us define the three types of ice creams as three sets:

• Let B be the set of people who eat Butterscotch ice cream.

• Let V be the set of people who eat Vanilla ice cream.

• Let C be the set of people who eat Chocolate ice cream.

From the problem statement, we are given the following details:

1. Total number of persons in the party = 210. Since all of them eat at least one type of ice cream, the total union of all three sets is 210:

$n(B\cup V\cup C)=210$

2. Number of people who eat only Butterscotch = 40.

3. Number of people who eat all three flavoured ice creams = 30.

$n(B\cap V\cap C)=30$

4. Total number of people who eat Butterscotch, $n\left(B\right)=130$.

5. Total number of people who eat Vanilla, $n\left(V\right)=100$.

6. Number of people who eat Butterscotch and Vanilla only = 40.

7. Number of people who eat Chocolate and Vanilla only = 10.

Step 1: Find the number of people who eat Butterscotch and Chocolate only

The set of people who eat Butterscotch, $n\left(B\right)$, consists of four mutually exclusive regions:

• Only Butterscotch

• Butterscotch and Vanilla only

• Butterscotch and Chocolate only

• All three flavours

Using the values given:

$n\left(B\right)=\text{(Only Butterscotch)}+\text{(Butterscotch \& Vanilla only)}+\text{(Butterscotch \& Chocolate only)}+\text{(All three)}$

$130=40+40+\text{(Butterscotch \& Chocolate only)}+30$

$130=110+\text{(Butterscotch \& Chocolate only)}$

$\text{Butterscotch \& Chocolate only}=130-110=20$

Step 2: Find the number of people who eat only Vanilla

Similarly, the set of people who eat Vanilla, $n\left(V\right)$, consists of four regions:

• Only Vanilla

• Butterscotch and Vanilla only

• Chocolate and Vanilla only

• All three flavours

Using the values given:

$n\left(V\right)=\text{(Only Vanilla)}+\text{(Butterscotch \& Vanilla only)}+\text{(Chocolate \& Vanilla only)}+\text{(All three)}$

$100=\text{(Only Vanilla)}+40+10+30$

$100=\text{(Only Vanilla)}+80$

$\text{Only Vanilla}=100-80=20$

Step 3: Find the number of people who eat only Chocolate

The total number of people in the party (210) is the sum of all 7 non-overlapping regions in a 3-set Venn diagram:

1. Only Butterscotch = 40

2. Only Vanilla = 20

3. Only Chocolate = ?

4. Butterscotch and Vanilla only = 40

5. Butterscotch and Chocolate only = 20

6. Chocolate and Vanilla only = 10

7. All three = 30

Adding all these regions together:

$210=40+20+\text{(Only Chocolate)}+40+20+10+30$

$210=160+\text{(Only Chocolate)}$

$\text{Only Chocolate}=210-160=50$

Therefore, the number of people who eat only Chocolate is 50.