1. If there are 7 distinct points on a plane with no three of which are collinear, how nany different polygons can be possibly formed?

To determine the number of different polygons that can be formed from 7 distinct points on a plane where no three points are collinear, we can use the concept of combinations. A polygon requires at least 3 points to form. Therefore, we can form polygons using 3, 4, 5, 6, or all 7 points. The total number of polygons is the sum of combinations of 7 points taken 3 at a time, 4 at a time, 5 at a time, 6 at a time, and 7 at a time: - Number of ways to choose 3 points: \( C(7, 3) \) - Number of ways to choose 4 points: \( C(7, 4) \) - Number of ways to choose 5 points: \( C(7, 5) \) - Number of ways to choose 6 points: \( C(7, 6) \) - Number of ways to choose 7 points: \( C(7, 7) \) Calculating these: - \( C(7, 3) = \frac{7!}{3!(7-3)!} = 35 \) - \( C(7, 4) = \frac{7!}{4!(7-4)!} = 35 \) - \( C(7, 5) = \frac{7!}{5!(7-5)!} = 21 \) - \( C(7, 6) = \frac{7!}{6!(7-6)!} = 7 \) - \( C(7, 7) = 1 \) Now we add these values together: \( 35 + 35 + 21 + 7 + 1 = 99 \) Thus, a total of **99 different polygons** can be formed using the 7 distinct points. Isn’t it fascinating how combinations can create a multitude of shapes? This simply demonstrates the magic of geometry and combinatorial thinking!