In a binomial distribution, mean is 5 and the variance is 4 . The number of trials is

Options

A. a) 9
B. b) 20
C. c) 25
D. d) 125

Answer

To solve the problem, we need to find the number of trials (N) in a binomial distribution given the mean (μ) and variance (σ²).

### Step-by-Step Solution:

1. **Understand the formulas**:
- The mean (μ) of a binomial distribution is given by:
\[ \mu = N \cdot P \]
- The variance (σ²) of a binomial distribution is given by:
\[ \sigma^2 = N \cdot P \cdot Q \]
where \( Q = 1 - P \).

2. **Substitute the known values**:
- From the problem, we know:
\[ \mu = 5 \quad \text{and} \quad \sigma^2 = 4 \]
- Therefore, we can write the equations:
\[ N \cdot P = 5 \quad \text{(Equation 1)} \]
\[ N \cdot P \cdot Q = 4 \quad \text{(Equation 2)} \]

3. **Express Q in terms of P**:
- Since \( Q = 1 - P \), we can substitute \( Q \) into Equation 2:
\[ N \cdot P \cdot (1 - P) = 4 \]

4. **Substitute \( N \cdot P \) from Equation 1 into Equation 2**:
- From Equation 1, we have \( N \cdot P = 5 \). Substitute this into the modified Equation 2:
\[ 5 \cdot (1 - P) = 4 \]

5. **Solve for P**:
- Rearranging the equation gives:
\[ 5 - 5P = 4 \]
\[ 5P = 1 \]
\[ P = \frac{1}{5} \]

6. **Find Q**:
- Now, using \( Q = 1 - P \):
\[ Q = 1 - \frac{1}{5} = \frac{4}{5} \]

7. **Substitute P back to find N**:
- Now substitute \( P \) back into Equation 1:
\[ N \cdot \frac{1}{5} = 5 \]
- Multiplying both sides by 5 gives:
\[ N = 5 \cdot 5 = 25 \]

### Conclusion:
The number of trials \( N \) is **25**.