In a group of students , 90 students know Hindi , 50 know English and 15 know both Each students knows either Hindi or English . How many students are there in the group ?

Answer

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

- \( H \): the set of students who know Hindi
- \( E \): the set of students who know English

We are given the following information:

- \( |H| = 90 \) (the number of students who know Hindi)
- \( |E| = 50 \) (the number of students who know English)
- \( |H \cap E| = 15 \) (the number of students who know both Hindi and English)

We need to find the total number of students in the group, which can be represented as \( |H \cup E| \) (the number of students who know either Hindi or English or both).

Using the formula for the union of two sets:

\[ |H \cup E| = |H| + |E| - |H \cap E| \]

Now, substituting the values we have:

\[ |H \cup E| = 90 + 50 - 15 \]

Calculating this step-by-step:

1. Add the number of students who know Hindi and English:
\[ 90 + 50 = 140 \]

2. Subtract the number of students who know both languages:
\[ 140 - 15 = 125 \]

Thus, the total number of students in the group is \( |H \cup E| = 125 \).

### Final Answer:
There are 125 students in the group.