In a group of 400 persons 250 can speak Hindi and 200 can speak English. Each person speak atleast one language. Find how many persons can speak both languages ?

Answer

To solve the problem, we will use the principle of inclusion-exclusion. Let's denote:

- \( N(H) \) = Number of people who speak Hindi = 250
- \( N(E) \) = Number of people who speak English = 200
- \( N(H \cup E) \) = Total number of people who speak at least one language = 400
- \( N(H \cap E) \) = Number of people who speak both languages (Hindi and English)

According to the principle of inclusion-exclusion, we have the formula:

\[ N(H \cup E) = N(H) + N(E) - N(H \cap E) \]

Now, we can substitute the known values into this formula:

\[ 400 = 250 + 200 - N(H \cap E) \]

Now, let's simplify the equation:

1. Combine the numbers on the right side:

\[ 400 = 450 - N(H \cap E) \]

2. Rearranging the equation to solve for \( N(H \cap E) \):

\[ N(H \cap E) = 450 - 400 \]

3. Calculate the value:

\[ N(H \cap E) = 50 \]

Thus, the number of persons who can speak both languages is **50**.