Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 15-16: Consider any eight points such that no three are collinear. How many triangles are determined?

Step-by-Step Solution

Answer

56 triangles can be formed.

Step 1: Understand the Problem

We have 8 points on a plane, where no three points are in a straight line (collinear). We need to determine how many unique triangles can be formed using these points.

Step 2: Identify Formula for Combinations

To find the number of triangles, we use the combination formula to choose 3 points from the 8 available points. The combination formula for choosing r items from n items is given by \( \binom{n}{r} \).

Step 3: Apply Combination Formula

Since we're choosing 3 points to form a triangle, we use \( n = 8 \) and \( r = 3 \). The formula is: \[ \binom{8}{3} = \frac{8!}{3!(8-3)!} \].

Step 4: Calculate Factorials

Calculate the required factorials: - \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)- \( 3! = 3 \times 2 \times 1 \)- \( (8-3)! = 5! = 5 \times 4 \times 3 \times 2 \times 1 \)

Step 5: Perform Division

Simplify the expression: \[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \] Calculate step by step: - Numerator: \( 8 \times 7 \times 6 = 336 \)- Denominator: \( 3 \times 2 \times 1 = 6 \)- Division: \( 336 \div 6 = 56 \)

Step 6: Conclusion

Therefore, the number of triangles that can be formed is 56.

Key Concepts

Triangles

Triangles are three-sided polygons, which means they have three edges and three vertices. Forming a triangle requires selecting three points that will act as the vertices of the triangle. In this particular problem, we have eight points on a plane, and no three of them are in a straight line, which is a condition that ensures each set of three points selected will form a triangle.
Understanding this is crucial because if three points were collinear (on a single straight line), they wouldn't form the expected triangle. Therefore, when considering combinatorics and triangle formation, always ensure that no three points are collinear to meet the triangle's criteria.
Triangles are fundamental figures in geometry and have properties like angles summing up to 180 degrees. They serve several purposes in mathematics, engineering, architecture, and many other fields.

Combination Formula

The combination formula is a mathematical way to determine the number of ways to choose a subset of items from a larger set. It is particularly useful in scenarios like determining how many triangles can be formed from a given set of points.
The general combination formula is represented as \( \binom{n}{r} \) and is read as \( n \) choose \( r \). Here, \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.
The formula is expressed as:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

The factorial symbol "!" represents the product of all positive integers up to that number. This formula helps us find different combinations efficiently without listing them all explicitly. This is crucial in fields like statistics and probability where such calculations are frequent.

Factorials

Factorials are an essential concept in combinatorics, especially when using the combination formula. The factorial of a non-negative integer \( n \) is denoted by \( n! \), and it's the product of all positive integers less than or equal to \( n \).
For example:

• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \)

Factorials grow very quickly with increasing \( n \), providing large numbers even for relatively small \( n \). They are a key component in calculating combinations and permutations, which are prevalent in solving problems involving arrangements and selections like forming triangles from a set of points.