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 (Option 4).

Let us analyze the given information step-by-step using set theory and Venn diagram concepts for three flavors of ice-cream: Butterscotch (B), Vanilla (V), and Chocolate (C).

Let:
- Total number of persons in the party, N(U) = 210
- Total people eating Butterscotch, n(B) = 130
- Total people eating Vanilla, n(V) = 100
- People eating all three flavors = n(B ∩ V ∩ C) = 30
- People eating ONLY Butterscotch = 40
- People eating Butterscotch and Vanilla ONLY = 40
- People eating Chocolate and Vanilla ONLY = 10

We are required to find how much less the number of people eating only Vanilla is compared to the number of people eating all three types of ice-creams.

First, let us examine the components that make up the total number of Vanilla eaters, n(V):

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

Substitute the given values into the equation:

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

Simplifying the right hand side:

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

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

So, the number of people who eat only Vanilla is 20.

Now, we compare this with the number of people eating all three types (which is 30):

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

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