In a group of 45 students, 22 can speak Hindi only and 12 can speak English only. If `(2lambda+1)` student can speak both Hindi and English, the value of `lambda` is

Explanation

To solve the problem step by step, we will use the information provided about the students who can speak Hindi and English.

### Step 1: Define the Sets
Let:
- \( U \) be the set of all students.
- \( H \) be the set of students who speak Hindi.
- \( E \) be the set of students who speak English.

### Step 2: Identify the Given Information
From the problem, we know:
- Total number of students \( |U| = 45 \)
- Students who speak Hindi only \( |H| = 22 \)
- Students who speak English only \( |E| = 12 \)
- Students who speak both Hindi and English \( |H \cap E| = 2\lambda + 1 \)

### Step 3: Set Up the Equation
The total number of students can be expressed as the sum of students who speak only Hindi, only English, and both languages:
\[ |U| = |H| + |E| + |H \cap E| \]
Substituting the known values:
\[ 45 = 22 + 12 + (2\lambda + 1) \]

### Step 4: Simplify the Equation
Combine the constants on the right side:
\[ 45 = 34 + (2\lambda + 1) \]
This simplifies to:
\[ 45 = 35 + 2\lambda \]

### Step 5: Solve for \( \lambda \)
Subtract 35 from both sides:
\[ 45 - 35 = 2\lambda \]
\[ 10 = 2\lambda \]
Now, divide both sides by 2:
\[ \lambda = \frac{10}{2} = 5 \]

### Final Answer
The value of \( \lambda \) is \( 5 \).

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