Solve the following problems independently of each other
Find the value of m, if there are maximum 28 triangles formed by joining total 9 points on a plane, out of which m points are collinear

Options

A. 3
B. 4
C. 1
D. 8

Answer

To solve the problem of finding the value of \( m \) such that there are a maximum of 28 triangles formed by joining a total of 9 points on a plane, out of which \( m \) points are collinear, we can follow these steps:

### Step 1: Understand the Problem
We have 9 points in total, and \( m \) of these points are collinear. The maximum number of triangles that can be formed by any 3 points from these 9 points is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of points and \( r \) is the number of points to choose (in this case, 3).

### Step 2: Calculate Total Triangles Without Collinearity
The total number of triangles that can be formed from 9 points is:
\[ \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

### Step 3: Calculate the Effect of Collinear Points
When \( m \) points are collinear, they cannot form a triangle. The number of triangles that can be formed from the \( m \) collinear points is:
\[ \binom{m}{3} \]
Thus, the maximum number of triangles that can be formed with the remaining points is:
\[ \text{Maximum triangles} = \binom{9}{3} - \binom{m}{3} \]

### Step 4: Set Up the Equation
According to the problem, the maximum number of triangles formed is 28. Therefore, we can set up the equation:
\[ 84 - \binom{m}{3} = 28 \]

### Step 5: Solve for \( \binom{m}{3} \)
Rearranging the equation gives:
\[ \binom{m}{3} = 84 - 28 = 56 \]

### Step 6: Find Possible Values of \( m \)
Now, we need to find \( m \) such that:
\[ \binom{m}{3} = 56 \]
This can be expressed as:
\[ \frac{m(m-1)(m-2)}{6} = 56 \]
Multiplying both sides by 6 gives:
\[ m(m-1)(m-2) = 336 \]

### Step 7: Test Possible Values of \( m \)
We can test integer values for \( m \):

- For \( m = 7 \):
\[ 7 \times 6 \times 5 = 210 \quad (\text{not } 336) \]

- For \( m = 8 \):
\[ 8 \times 7 \times 6 = 336 \quad (\text{correct!}) \]

- For \( m = 9 \):
\[ 9 \times 8 \times 7 = 504 \quad (\text{too high}) \]

### Conclusion
The only value of \( m \) that satisfies the equation is \( m = 8 \).

### Final Answer
Thus, the value of \( m \) is \( \boxed{8} \).