There are 12 points in a plane of which 5 are collinear. The maximum number of distinct quadrilaterals which can be formed with vertices at these points is:

Options

A. `2.""^(7)O_(3)`
B. `""^(7)P_(3)`
C. `10*""^(7)C_(3)`
D. `420`

Answer

To solve the problem of finding the maximum number of distinct quadrilaterals that can be formed with 12 points in a plane, of which 5 are collinear, we can follow these steps:

### Step 1: Understand the Problem
We have a total of 12 points, out of which 5 points are collinear. A quadrilateral requires 4 vertices. However, if all 4 vertices are chosen from the 5 collinear points, they will not form a quadrilateral.

### Step 2: Calculate Total Combinations
First, we calculate the total number of ways to choose 4 points from the 12 points without considering the collinearity:
\[ \text{Total combinations} = \binom{12}{4} \]
Using the formula for combinations:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
we have:
\[ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]

### Step 3: Subtract Invalid Combinations
Next, we need to subtract the combinations that consist of all 4 points being chosen from the 5 collinear points:
\[ \text{Invalid combinations} = \binom{5}{4} \]
Calculating this gives:
\[ \binom{5}{4} = 5 \]

### Step 4: Calculate Valid Combinations
Now, we subtract the invalid combinations from the total combinations:
\[ \text{Valid combinations} = \text{Total combinations} - \text{Invalid combinations} = 495 - 5 = 490 \]

### Step 5: Conclusion
Thus, the maximum number of distinct quadrilaterals that can be formed with vertices at these points is:
\[ \boxed{490} \]