in a class of 45 student, 22 can speak hindi and 12 can speak English only . The number of students , who can speak both Hindi and English , is

Options

A. 9
B. 11
C. 23
D. 17

Answer

To solve the problem step by step, we can use the principle of inclusion-exclusion for sets.

### Step 1: Define the variables
Let:
- \( N_H \) = Number of students who can speak Hindi = 22
- \( N_E \) = Number of students who can speak English = 12 (only English)
- \( X \) = Number of students who can speak both Hindi and English

### Step 2: Write down the total number of students
We know the total number of students in the class is 45. Therefore, we can express this as:
\[ N_H \cup N_E = 45 \]

### Step 3: Express the total number of students in terms of \( X \)
According to the principle of inclusion-exclusion:
\[ N_H \cup N_E = N_H + N_E - N_H \cap N_E \]
Where \( N_H \cap N_E \) is the number of students who can speak both languages, which we have defined as \( X \).

Substituting the values we have:
\[ 45 = (22 + X) + (12 + X) - X \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ 45 = 22 + X + 12 + X - X \]
\[ 45 = 22 + 12 + X \]
\[ 45 = 34 + X \]

### Step 5: Solve for \( X \)
Now, isolate \( X \):
\[ X = 45 - 34 \]
\[ X = 11 \]

### Conclusion
The number of students who can speak both Hindi and English is \( \boxed{11} \).

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