Find a positive value of m for which the coefficient of `x^2` in the expansion of `(1+x)^m` is 6.

Explanation

To find a positive value of \( m \) for which the coefficient of \( x^2 \) in the expansion of \( (1+x)^m \) is 6, we can follow these steps:

### Step 1: Identify the Coefficient of \( x^2 \)
The coefficient of \( x^2 \) in the expansion of \( (1+x)^m \) can be expressed using the binomial coefficient:
\[ \binom{m}{2} = \frac{m(m-1)}{2} \]
We want this coefficient to equal 6:
\[ \binom{m}{2} = 6 \]

### Step 2: Set Up the Equation
From the above expression, we can set up the equation:
\[ \frac{m(m-1)}{2} = 6 \]

### Step 3: Eliminate the Fraction
To eliminate the fraction, multiply both sides by 2:
\[ m(m-1) = 12 \]

### Step 4: Rearrange the Equation
Rearranging gives us a standard quadratic equation:
\[ m^2 - m - 12 = 0 \]

### Step 5: Factor the Quadratic Equation
Next, we will factor the quadratic equation:
\[ m^2 - 4m + 3m - 12 = 0 \]
This can be factored as:
\[ (m - 4)(m + 3) = 0 \]

### Step 6: Solve for \( m \)
Setting each factor to zero gives us:
\[ m - 4 = 0 \quad \text{or} \quad m + 3 = 0 \]
Thus, we find:
\[ m = 4 \quad \text{or} \quad m = -3 \]

### Step 7: Select the Positive Value
Since we are looking for a positive value of \( m \), we select:
\[ m = 4 \]

### Final Answer
The positive value of \( m \) for which the coefficient of \( x^2 \) in the expansion of \( (1+x)^m \) is 6 is:
\[ \boxed{4} \]