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 1: Identify the Coefficient of \( x \)
The coefficient of \( x \) in the expansion can be found by considering the first-order terms from both expansions:
- From \( (1 + x)^m \), the coefficient of \( x \) is \( \binom{m}{1} = m \).
- From \( (1 - x)^n \), the coefficient of \( x \) is \( \binom{n}{1}(-1) = -n \).

Thus, the total coefficient of \( x \) is:
\[ m - n = 3 \quad \text{(Equation 1)} \]

### Step 2: Identify the Coefficient of \( x^2 \)
The coefficient of \( x^2 \) can be found by considering the second-order terms:
- From \( (1 + x)^m \), the coefficient of \( x^2 \) is \( \binom{m}{2} = \frac{m(m-1)}{2} \).
- From \( (1 - x)^n \), the coefficient of \( x^2 \) is \( \binom{n}{2}(-1) = -\frac{n(n-1)}{2} \).

Thus, the total coefficient of \( x^2 \) is:
\[ \frac{m(m-1)}{2} - \frac{n(n-1)}{2} = -6 \]
Multiplying through by 2 to eliminate the fraction gives:
\[ m(m-1) - n(n-1) = -12 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
Expanding Equation 2:
\[ m^2 - m - n^2 + n = -12 \]
Rearranging gives:
\[ m^2 - n^2 - m + n + 12 = 0 \]

### Step 4: Factor the Expression
Notice that \( m^2 - n^2 \) can be factored as:
\[ (m - n)(m + n) - (m - n) + 12 = 0 \]
Factoring out \( (m - n) \):
\[ (m - n)(m + n - 1) + 12 = 0 \]

### Step 5: Substitute from Equation 1
From Equation 1, we know \( m - n = 3 \). Substituting this into the equation gives:
\[ 3(m + n - 1) + 12 = 0 \]
Simplifying:
\[ 3(m + n - 1) = -12 \quad \Rightarrow \quad m + n - 1 = -4 \quad \Rightarrow \quad m + n = -3 \quad \text{(Equation 3)} \]

### Step 6: Solve the System of Equations
Now we have two equations:
1. \( m - n = 3 \)
2. \( m + n = -3 \)

Adding these two equations:
\[ (m - n) + (m + n) = 3 - 3 \quad \Rightarrow \quad 2m = 0 \quad \Rightarrow \quad m = 0 \]

Substituting \( m = 0 \) into Equation 1:
\[ 0 - n = 3 \quad \Rightarrow \quad n = -3 \]

### Final Values
Thus, the values of \( m \) and \( n \) are:
\[ m = 0, \quad n = -3 \]