Question

In a group of 100 students, 85 students speak Tamil. 40 students speak English,
20 students speak French, 32 students speak Tamil and English, 13 speak English
and French and 10 speak Tamil and French. If each student knows atleast any
one of these languages then find the number of students who speak all these
three languages.​

Answer 1

$\mathtt{ \huge{ \fbox{SOLUTION :}}}$

Let , T , E and F be the sets of students who knows Tamil , English and French

Accordingly ,

$\sf \mapsto n(T) = 85 \\ \\ \sf \mapsto n(E) = 40\\ \\ \sf \mapsto n(F) = 20\\ \\ \sf \mapsto n(T \cap E) = 32\\ \\ \sf \mapsto n(E \cap F) = 13\\ \\ \sf \mapsto n(F \cap T) = 10 \\ \\ \sf \mapsto n(T \cup E \cup F ) = 100$

We know that ,

$\large \sf \fbox{n(A \cup B \cup C ) = n(A) + n(B) + n(C) - n(A \cap B ) - n(B \cap C) - n(C \cap A )}$

So , Substitute the known values , we obtain

$\sf \hookrightarrow 100 = 85 + 40 + 20 - 32 - 13 - 10 + n(T \cap E \cap F ) \\ \\ \sf \hookrightarrow 100 = 145 - 55 + n(T \cap E \cap F ) \\ \\ \sf \hookrightarrow n(T \cap E \cap F ) = 100 - 90 \\ \\ \sf \hookrightarrow n(T \cap E \cap F ) = 10$

Hence , 10 is the number students who speak all these three languages

Answer 2

Answer:

Hope you got the correct answer