If in the expansion of ` (1 +x)^(m) (1 - x)^(n)` , the coefficients
of x and ` x^(2) ` are 3 and - 6 respectively, the value of m and n are

Options

A. 3
B. 6
C. 9
D. 12

Explanation

To solve the problem, we need to find the values of \( m \) and \( n \) given that the coefficients of \( x \) and \( x^2 \) in the expansion of \( (1 + x)^m (1 - x)^n \) are 3 and -6 respectively.

### Step-by-Step Solution:

1. **Identify the Coefficient of \( x \)**:
The coefficient of \( x \) in the expansion of \( (1 + x)^m (1 - x)^n \) can be found using the binomial theorem:
\[ \text{Coefficient of } x = \binom{m}{1} \cdot \binom{n}{0} + \binom{m}{0} \cdot \binom{n}{1} (-1) = m - n \]
Given that this coefficient equals 3, we have:
\[ m - n = 3 \quad \text{(Equation 1)} \]

2. **Identify the Coefficient of \( x^2 \)**:
The coefficient of \( x^2 \) can be calculated similarly:
\[ \text{Coefficient of } x^2 = \binom{m}{2} \cdot \binom{n}{0} + \binom{m}{1} \cdot \binom{n}{1} (-1) + \binom{m}{0} \cdot \binom{n}{2} = \frac{m(m-1)}{2} - mn + \frac{n(n-1)}{2} \]
Given that this coefficient equals -6, we have:
\[ \frac{m(m-1)}{2} - mn + \frac{n(n-1)}{2} = -6 \quad \text{(Equation 2)} \]

3. **Multiply Equation 2 by 2 to eliminate the fraction**:
\[ m(m-1) - 2mn + n(n-1) = -12 \]
Rearranging gives:
\[ m^2 - m + n^2 - n - 2mn = -12 \]

4. **Rearranging the equation**:
We can rearrange this to:
\[ m^2 - 2mn + n^2 - m - n + 12 = 0 \]
This can be rewritten as:
\[ (m - n)^2 - (m + n) + 12 = 0 \]

5. **Substituting \( m - n = 3 \) from Equation 1**:
Substitute \( m - n = 3 \) into the equation:
\[ 3^2 - (m + n) + 12 = 0 \]
Simplifying gives:
\[ 9 - (m + n) + 12 = 0 \implies m + n = 21 \quad \text{(Equation 3)} \]

6. **Solving the system of equations**:
Now we have two equations:
- \( m - n = 3 \) (Equation 1)
- \( m + n = 21 \) (Equation 3)

Adding these two equations:
\[ (m - n) + (m + n) = 3 + 21 \implies 2m = 24 \implies m = 12 \]

Substituting \( m = 12 \) back into Equation 1:
\[ 12 - n = 3 \implies n = 12 - 3 = 9 \]

### Final Answer:
Thus, the values of \( m \) and \( n \) are:
\[ m = 12, \quad n = 9 \]