How Many Straight Lines Can Be Formed from 15 Non-Collinear Points?

When dealing with a set of points in a plane, one of the fundamental questions in combinatorial geometry is determining how many straight lines can be formed by connecting pairs of these points. This problem can be elegantly solved using the concept of combinations.

Understanding the Problem

A straight line is uniquely determined by any two points. Therefore, if we have a set of 15 non-collinear points, we need to find the number of unique pairs of points that can be formed. The term non-collinear means that no three of these points lie on the same straight line, ensuring that every pair of points indeed forms a unique straight line.

Using Combinations

The mathematical solution to this problem involves a combination. A combination is a selection of items from a larger set, where the order of selection does not matter. The formula for combinations is given by:

Binom{n}{r}  frac{n!}{r!(n-r)!}

Here, n represents the total number of points, and r is the number of points to be chosen to form a line. In our case, n 15 and r 2, since a line is defined by any two points.

Calculating the Number of Lines

Let's substitute the values into the combination formula:

Binom{15}{2}  frac{15!}{2!15! - 2!}  frac{15 times 14}{2 times 1}  frac{210}{2}  105

This calculation shows that 105 unique lines can be formed from 15 non-collinear points.

Example and Confirmation

Let's consider a similar example with 16 non-collinear points. Using the same method, we can apply the combination formula:

Binom{16}{2}  frac{16!}{2!16! - 2!}  frac{16 times 15}{2 times 1}  120

Similarly, for three points to form a line, we use:

Binom{3}{2}  frac{3!}{2!3! - 2!}  3

Adding this to the previous 105 lines formed by 15 points, we can see that the principle of combinations holds true.

Conclusion

Understanding how many straight lines can be formed from a given set of non-collinear points is a crucial concept in geometry and has applications in various fields such as computer graphics, data science, and engineering. By using the combination formula, we can accurately determine the number of lines formed.

Remember, for any n non-collinear points, the number of lines that can be formed is:

Binom{n}{2}  frac{n(n-1)}{2}

This formula will always give the correct number of lines.