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.

To find the number of people who eat only Chocolate icecream, we can break down the problem using Venn diagram regions for the three flavours: Butterscotch (B), Vanilla (V), and Chocolate (C).

Let us list the given information:

- Total number of persons in the party = 210
- People who eat only Butterscotch = 40
- People who eat all three flavours = 30
- Total people who eat Butterscotch = 130
- Total people who eat Vanilla = 100
- People who eat Butterscotch and Vanilla only = 40
- People who eat Chocolate and Vanilla only = 10

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

The total number of people who eat Butterscotch is made up of four non-overlapping regions:

$\left(\mathrm{Only\; Butterscotch}\right)+\left(\mathrm{Butterscotch\; and\; Vanilla\; only}\right)+\left(\mathrm{Butterscotch\; and\; Chocolate\; only}\right)+\left(\mathrm{All\; three}\right)=130$

Substituting the known values into the equation:

$40+40+\left(\mathrm{Butterscotch\; and\; Chocolate\; only}\right)+30=130$

$110+\left(\mathrm{Butterscotch\; and\; Chocolate\; only}\right)=130$

$\mathrm{Butterscotch\; and\; Chocolate\; only}=130-110=20$

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

The total number of people who eat Vanilla is also made up of four non-overlapping regions:

$\left(\mathrm{Only\; Vanilla}\right)+\left(\mathrm{Butterscotch\; and\; Vanilla\; only}\right)+\left(\mathrm{Chocolate\; and\; Vanilla\; only}\right)+\left(\mathrm{All\; three}\right)=100$

Substituting the known values into the equation:

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

$\left(\mathrm{Only\; Vanilla}\right)+80=100$

$\mathrm{Only\; Vanilla}=100-80=20$

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

Since everyone at the party eats at least one type of icecream, the total number of people (210) is the sum of all 7 individual regions of the Venn diagram:

$\left(\mathrm{Only\; Butterscotch}\right)+\left(\mathrm{Only\; Vanilla}\right)+\left(\mathrm{Only\; Chocolate}\right)+\left(\mathrm{Butterscotch\; and\; Vanilla\; only}\right)+\left(\mathrm{Butterscotch\; and\; Chocolate\; only}\right)+\left(\mathrm{Chocolate\; and\; Vanilla\; only}\right)+\left(\mathrm{All\; three}\right)=210$

Substituting all the values calculated so far:

$40+20+\left(\mathrm{Only\; Chocolate}\right)+40+20+10+30=210$

$160+\left(\mathrm{Only\; Chocolate}\right)=210$

$\mathrm{Only\; Chocolate}=210-160=50$

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