In a binomial distribution `B(n, p),` the sum and the product of the mean and the variance are `5 `and `6 `respectively, then `6(n + p – q)` is equal to

Options

A. `50`
B. `53`
C. `52`
D. `51`

Answer

To solve the problem, we need to analyze the given conditions about the mean and variance of a binomial distribution \( B(n, p) \).

### Step-by-Step Solution:

1. **Understanding Mean and Variance**:
- The mean \( \mu \) of a binomial distribution is given by:
\[ \mu = np \]
- The variance \( \sigma^2 \) of a binomial distribution is given by:
\[ \sigma^2 = npq \]
where \( q = 1 - p \).

2. **Setting Up the Equations**:
- We are given that the sum of the mean and variance is 5:
\[ np + npq = 5 \]
- We are also given that the product of the mean and variance is 6:
\[ np \cdot npq = 6 \]
- Let \( x = np \) and \( y = npq \). Thus, we have:
\[ x + y = 5 \quad \text{(1)} \]
\[ xy = 6 \quad \text{(2)} \]

3. **Finding \( x \) and \( y \)**:
- From equation (1), we can express \( y \) in terms of \( x \):
\[ y = 5 - x \]
- Substituting this into equation (2):
\[ x(5 - x) = 6 \]
\[ 5x - x^2 = 6 \]
\[ x^2 - 5x + 6 = 0 \]
- Factoring the quadratic:
\[ (x - 2)(x - 3) = 0 \]
- Thus, \( x = 2 \) or \( x = 3 \).

4. **Determining Corresponding \( y \)**:
- If \( x = 2 \):
\[ y = 5 - 2 = 3 \]
- If \( x = 3 \):
\[ y = 5 - 3 = 2 \]

5. **Finding \( p \) and \( q \)**:
- For \( x = 2 \) (i.e., \( np = 2 \)) and \( y = 3 \) (i.e., \( npq = 3 \)):
\[ q = \frac{y}{x} = \frac{3}{2} \quad \text{(not possible since \( q \) must be } \leq 1) \]
- For \( x = 3 \) (i.e., \( np = 3 \)) and \( y = 2 \) (i.e., \( npq = 2 \)):
\[ q = \frac{y}{x} = \frac{2}{3} \]
\[ p = 1 - q = 1 - \frac{2}{3} = \frac{1}{3} \]

6. **Finding \( n \)**:
- From \( np = 3 \):
\[ n \cdot \frac{1}{3} = 3 \implies n = 9 \]

7. **Calculating \( 6(n + p - q) \)**:
- Now we need to find \( 6(n + p - q) \):
\[ n + p - q = 9 + \frac{1}{3} - \frac{2}{3} = 9 - \frac{1}{3} = \frac{27}{3} - \frac{1}{3} = \frac{26}{3} \]
- Therefore:
\[ 6(n + p - q) = 6 \cdot \frac{26}{3} = \frac{156}{3} = 52 \]

### Final Answer:
\[ 6(n + p - q) = 52 \]