Question

In a class of 50 students, \(23\) speak English (E), \(15\) Hindi (H), \(18\) Punjabi (P). Only \(E&H=3\), only \(H&P=6\), only \(E&P=6\). If \(9\) speak only English, how many speak all three languages?

Hint:

When several “only” region counts are given, use the total of a set (e.g., \(|E|\)) to solve directly for the triple intersection.

Answer 1

Let \(x=|E\cap H\cap P|\). Using \(|E|=23\): \[ |E|=\text{only }E+\text{only }(E&H)+\text{only }(E&P)+x =9+3+6+x=18+x. \] Thus \(23=18+x \Rightarrow x=5\). (Totals for \(H\) and \(P\) then give \(H\text{-only}=1\), \(P\text{-only}=1\), consistent.) \[ \boxed{5} \]