Calculating the Number of Lines Formed by Joining Pairs of Points: A Geometric Perspective

When considering a set of points on a plane, a common geometric challenge is determining the number of lines that can be formed by joining pairs of points. Specifically, given a set of points, where no three points are collinear, we need to determine how many such lines can be formed.

Using Combinations to Solve the Problem

A systematic approach to this problem is to use combinatorial mathematics, specifically the concept of combinations. The formula for combinations is given by:

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

Here, n is the total number of points, and r is the number of points chosen to form a line. In the context of our problem, n 5 and r 2. Plugging these values into the formula, we calculate:

binom{5}{2} frac{5!}{2!(5-2)!} frac{5 times 4}{2 times 1} 10

Hence, the number of lines that can be formed by joining the 5 points is 10.

Combinatorial Interpretation

The combinatorial approach can also be explained through a more detailed step-by-step method. If we have 5 points and wish to form lines by connecting pairs of these points, we can start with one point and connect it to all the other points. Each of the remaining 4 points will form a unique line with the starting point. Since there are 5 points, we initially count each point forming 4 lines, giving us a total of 5 times 4 lines.

However, this method counts each line twice (once for each end of the line). Therefore, to get the correct number of unique lines, we divide the total by 2:

5 times 4 / 2 10

Generalization for Any Number of Points

The above method can be generalized for any number of points n. The number of lines can be expressed as:

n(n - 1) / 2

This formula works because each point can be connected to n - 1 other points. Since each line is counted twice, we divide by 2 to obtain the number of unique lines.

Example: 5 Points

For 5 points, using the general formula:

5(5 - 1) / 2 5 times 4 / 2 10

We can also verify this step-by-step:

The first point connects to 4 others (4 lines). The second point connects to 3 new points (4 3 7 lines). The third point connects to 2 new points (7 2 9 lines). The fourth point connects to 1 new point (9 1 10 lines). The fifth point does not introduce any new lines (10 lines total).

Conclusion

The problem of determining the number of lines formed by joining pairs of points, where no three points are collinear, can be elegantly solved using combinatorial methods. Whether through the direct application of the combination formula or the more detailed step-by-step approach, the result is consistently 10 lines for 5 points. This method can be generalized for any number of points, providing a powerful tool for solving similar geometric problems.