There are 210 persons in a party, and all of them eat different 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. 40 people eat butterscotch and vanila only, 10 people eat chocolate and vanila only.

Number of people eating only vanilla is how much less than the people eating all three types of ice-creams?

The correct answer is 10.

Step-by-Step Explanation:

Let us denote the three types of ice-cream flavors as follows:
• B: Butterscotch
• V: Vanilla
• C: Chocolate

From the given problem, we have the following data:
• Total number of people eating Vanilla, $n\left(V\right)=100$
• Number of people eating Butterscotch and Vanilla only = 40
• Number of people eating Chocolate and Vanilla only = 10
• Number of people eating all three flavors, $n(B\cap V\cap C)=30$

Step 1: Calculate the number of people eating only Vanilla

The total set of people who eat Vanilla is the sum of four disjoint regions:
1. People eating only Vanilla
2. People eating Butterscotch and Vanilla only
3. People eating Chocolate and Vanilla only
4. People eating all three flavors

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

Substituting the given values into the equation:

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

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

$n\left(\text{Only Vanilla}\right)=100-80=20$

So, 20 people eat only Vanilla.

Step 2: Find the difference between people eating all three flavors and people eating only Vanilla

$\text{Difference}=n\left(\text{All three}\right)-n\left(\text{Only Vanilla}\right)$

$\text{Difference}=30-20=10$

Therefore, the number of people eating only vanilla is 10 less than the number of people eating all three types of ice-creams.