Combinations

Learning Objectives Define a combination and its formula in a geometric context. Apply the combination formula to calculate the number of angles that can be formed from a set of non-collinear points. Calculate the number of triangles that can be formed from the vertices of a polygon. Differentiate between problems requiring combinations and those requiring permutations in geometric counting. Solve problems involving combinations to find the number of specific types of triangles (e.g., right-angled) based on angle properties. Analyze geometric configurations to determine the number of possible diagonals or lines, which in turn define angles. How many different triangles can you draw using the vertices of a dodecagon (12-sided shape)? 🤯 Let's find out without drawing a s...

TermDefinitionExample CombinationA selection of items from a set where the order of selection does not matter.Choosing 3 vertices {A, B, C} from a pentagon to form a triangle is a combination. The triangle ABC is the same as triangle BCA or CAB. VertexA point where two or more lines, rays, or edges meet. These are the points we select to form angles and polygons.In a square ABCD, the points A, B, C, and D are its vertices. AngleA figure formed by two rays sharing a common endpoint, called the vertex of the angle. To define an angle, three points are needed.Given points P, O, and Q, the angle ∠POQ is formed with O as the vertex. The order matters here, as ∠POQ is different from ∠OPQ. PolygonA closed plane figure with three or more straight sides and angles. We often use the vertices of a p...

The Combination Formula C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!} Use this formula to find the number of ways to choose k items from a set of n distinct items where order does not matter. 'n' is the total number of items to choose from (e.g., vertices), and 'k' is the number of items being chosen (e.g., 3 for a triangle). Counting Triangles from Polygon Vertices Number of triangles = C(n, 3) A triangle is defined by 3 non-collinear vertices. For a convex polygon with 'n' vertices, any choice of 3 vertices will form a unique triangle. Since the order of choosing vertices doesn't matter, we use combinations. Counting Angles from a Set of Points Number of angles = n \times C(n-1, 2) To form an angle, you need a specific vertex and t...