Question

10 distinct points are taken on a circle. Then using these points
Statement I : The number of triangles that can be formed is 100
Statement II : The number of chords that can be formed is 45
Which of the following is correct?

• A. Both Statement I and Statement II are true
• B. Both Statement I and Statement II are false
• C. Statement I is true and Statement II is false
• D. Statement I is false and Statement II is true

Hint:

The phrase "points on a circle" is a key indicator that the points are in a "general position," ensuring that no three points are collinear. This simplifies geometric combinatorics, allowing you to use simple combinations without worrying about subtracting degenerate cases.

Answer 1

The Correct Option is D

Let's evaluate the two statements based on the given problem about 10 distinct points on a circle.

1. Statement I: The number of triangles that can be formed is 100.
   • To form a triangle, we need to choose 3 points from the 10 distinct points on the circle.
   • The number of ways to choose 3 points out of 10 is given by the combination formula: \(^{10}C_3 = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120.\)
   • Therefore, the number of triangles that can be formed is 120, not 100. Thus, Statement I is false.
2. Statement II: The number of chords that can be formed is 45.
   • A chord is formed by connecting any two distinct points on the circle.
   • The number of ways to choose 2 points from 10 is given by the combination formula: \(^{10}C_2 = \frac{10 \times 9}{2 \times 1} = 45.\)
   • Therefore, the number of chords that can be formed is indeed 45. Thus, Statement II is true.

Given this analysis, the correct answer is: Statement I is false and Statement II is true.