A class has $175$ students. The following description gives the number of students one or more of the subjects in this class: mathematics $100,$ physics $70,$ chemistry $46,$ mathematics and physics $30 ;$ mathematics and chemistry $28 ;$ physics and chemistry $23 ;$ mathematics, physics and chemistry $18$. Find the number of students who have not offered any of these subjects.

 

Let $\underset{i=1}{\overset{50}{\cup }}{X}_i=\underset{i=1}{\overset{n}{\cup }}{Y}_i=T$, where each $X_i$ contains $10$ elements and each $Y_i$ contains $5$ elements. If each element of the set $T$ is an element of exactly $20$ of sets $X_i$'s and exactly $6$ of sets $Y_i$'s then $n$ is equal to :

In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?

Let $S_1, S_2$ and $S_3$ be three sets defined as

$S_1=\{z\in ℂ:|z-1|\le \sqrt{2}\},$

$S_2=\{z\in ℂ:Re((1-i)z)\ge 1\}$ and

$S_3=\{z\in ℂ:Im(z)\le 1\}.$

Then, the set $S_1\cap S_2\cap S_3$