Problem 74 - Algebra and Trigonometry

Question

55–75 Solve the problem using the appropriate counting principle(s). Geometry Twelve dots are drawn on a page in such a way that no three are collinear. How many straight lines can be formed by joining the dots?

Step-by-Step Solution

Answer

There are 66 lines possible.

Step 1: Understanding the Problem

The problem involves determining the number of lines that can be formed by joining two dots when there are twelve dots on a page. The restriction is that no three dots are collinear, meaning each combination of two dots will form a unique line.

Step 2: Applying the Counting Principle: Combinations

To find the number of lines, we need to select 2 dots out of 12 to form each line. This is a combination problem, calculated as \( \binom{n}{r} \), where \( n \) is the total number of dots and \( r \) is the number of dots needed for each line.

Step 3: Calculating the Combination

Calculate \( \binom{12}{2} \), which represents the number of ways to choose 2 dots from 12. This is calculated as \( \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \).

Step 4: Conclusion

Thus, the number of distinct lines that can be formed by joining these twelve dots is 66.

Key Concepts

Counting Principles

Counting principles are fundamental techniques in combinatorics that allow us to determine the number of ways an event can occur. In problems involving counting, we often use principles such as permutations and combinations. These tools help us break down complex counting situations into more manageable parts.

In the given exercise, we need to calculate the number of ways to form lines by selecting pairs of dots. Since the order in which we choose the dots does not matter (a line joining dot A to dot B is the same as joining dot B to dot A), we use combinations, not permutations.

• Permutations are used when the order does matter.
• Combinations are used when the order does not matter.

Understanding the appropriate counting principle to apply in a given situation is crucial for finding the correct solution.

Geometry

In geometry, straight lines are fundamental elements that connect two points. In the context of the exercise, 12 dots are placed on a page, and each pair of points can form a unique straight line if no three points are collinear.

This setting is crucial because it ensures each pair of points contributes to a distinct line. Additionally, it avoids scenarios where three or more points lie on the same line, which would reduce the total number of distinct lines formed.

Geometry not only provides the setting for this problem but also influences the counting. By assuring that each line is distinct, the problem emphasizes the uniqueness of each line formed. Thus, geometry sets the rules for how we apply counting principles.

Combinations

Combinations are used to calculate the number of ways to choose a subset of items from a larger set without regard to the order of selection. This is exactly what is needed in our problem. We are forming lines by choosing two dots out of 12.

The notation for combinations is \({\binom{n}{r}}\), which indicates the number of ways to choose \({r}\) items from \({n}\) items. This is calculated using the formula:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]For our problem, \({n=12}\) and \({r=2}\), giving us:\[ \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \]This calculation shows how many distinct lines can be formed.

Collinear Points

Collinear points are points that lie on the same straight line. In the exercise, it's critical to know that no three points among the twelve are collinear.

If some points were collinear, it would mean they lie on the same line, reducing the total number of unique lines possible. Since the problem states no three points are collinear, each pair of points forms a distinct line without sharing any other points on that line.

This condition simplifies the problem as it allows each pair selection to be considered independently when forming lines. Thus, while calculating combinations, as in this case, collinearity is a vital consideration for ensuring the accuracy of the total count.