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: Understand the Binomial Expansion
The binomial expansion of \( (1+x)^m \) can be expressed using the binomial theorem:
\[ (1+x)^m = \sum_{r=0}^{m} \binom{m}{r} x^r \]
where \( \binom{m}{r} \) is the binomial coefficient.

### Step 2: Identify the Coefficient of \( x^2 \)
In the expansion, the coefficient of \( x^2 \) corresponds to the term where \( r = 2 \). Thus, the coefficient of \( x^2 \) is given by:
\[ \binom{m}{2} = \frac{m(m-1)}{2} \]

### Step 3: Set Up the Equation
According to the problem, we want this coefficient to equal 6:
\[ \frac{m(m-1)}{2} = 6 \]

### Step 4: Solve for \( m \)
Multiply both sides by 2 to eliminate the fraction:
\[ m(m-1) = 12 \]
Rearranging gives us:
\[ m^2 - m - 12 = 0 \]

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

### Step 6: Find the Roots
Setting each factor to zero gives us the possible values for \( m \):
\[ m - 4 = 0 \quad \Rightarrow \quad m = 4 \]
\[ m + 3 = 0 \quad \Rightarrow \quad m = -3 \]

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

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