Question

In a committee, 50 people speak Hindi, 20 speak English and 10 speak both Hindi and English. How many speak at least one of these two languages?

Answer 1

Hint: Use the formula of set theory for the number of items $ n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) $ for any two items $ A $ and $ B $ . Also note that the number of items for each type is given in the question which is only required for solution.

Complete step-by-step answer:
Lets say H be the set of people who speak Hindi in the committee and lets say E be the set of people who speak English in the committee.
Number of people who speak Hindi = $ n(H) = 50 $
Number of people who speak English = \[n\left( E \right){\text{ }} = {\text{ }}20\]
Number of people who speak both Hindi and English = \[n(H \cap E){\text{ }} = 10\]
People who speak at least one language \[ = {\text{ }}n(H \cup E)\] \[ = {\text{ }}n(H \cup E)\]
\[\Rightarrow n\left( {H \cup E} \right){\text{ }} = {\text{ }}n\left( H \right) + n\left( E \right) - n\left( {H \cap E} \right)\]
\[n\left( {H \cup E} \right) = 50 + 20 - 10 = 60\]
Hence $ 60 $ people in the committee at least one of the two languages.

Note: If we consider two sets of people, one that speaks English and one that Hindi. So, the intersection of both the sets gives the persons that can speak both English and Hindi and the union of both the sets gives the persons that can speak at least one language which is required.