Question

• A. The quantity on the left is greater
• B. The quantity on the right is greater
• C. Both are equal
• D. The relationship cannot be determined without further information

Hint:

Problems involving selecting points to form geometric shapes (lines, triangles, polygons) from a set of non-collinear points are applications of combinations. A line/chord needs 2 points, a triangle needs 3, a quadrilateral needs 4, and so on.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem involves geometric figures formed by selecting points on a circle. Since the order in which we choose the points does not matter for forming a chord or a quadrilateral, these are combination problems.
Step 2: Key Formula or Approach:
The number of ways to choose \(r\) items from a set of \(n\) distinct items is given by the combination formula: \(C(n, r) = \frac{n!}{r!(n-r)!}\).
We also use the identity \(C(n, r) = C(n, n-r)\).
Step 3: Detailed Explanation:
For Column A:
There are 6 points on a circle. A chord is a line segment that connects two distinct points on the circle. To find the number of chords, we need to find the number of ways to choose 2 points from the 6 available points. \[ \text{Number of chords} = C(6, 2) = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \] So, Quantity A is 15.
For Column B:
There are 6 points on a circle. A quadrilateral is a polygon with four vertices. To form a quadrilateral, we need to choose 4 points from the 6 available points. \[ \text{Number of quadrilaterals} = C(6, 4) \] Using the identity \(C(n, r) = C(n, n-r)\): \[ C(6, 4) = C(6, 6-4) = C(6, 2) = \frac{6 \times 5}{2 \times 1} = 15 \] So, Quantity B is 15.
Step 4: Final Answer:
Comparing the two quantities:
Quantity A = 15
Quantity B = 15
The two quantities are equal.