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

Step 1: Understanding the Concept:
This problem applies combinations to a geometric setting. A triangle is uniquely determined by selecting any 3 non-collinear points. A chord is uniquely determined by selecting any 2 distinct points. Since all given points lie on a circle, no three points can be collinear.

Step 2: Key Formula or Approach:
- The number of ways to choose $k$ items from a set of $n$ distinct items is given by combinations: ${}^nC_k = \frac{n!}{k!(n-k)!}$. - To form a triangle, we need to choose 3 points out of the total $n$ points. So, number of triangles $= {}^nC_3$. - To form a chord, we need to connect 2 points out of the total $n$ points. So, number of chords $= {}^nC_2$.

Step 3: Detailed Explanation:
We are given $n = 10$ distinct points on a circle. Let's evaluate Statement I: A triangle requires 3 vertices. Since no three points on a circle are collinear, any combination of 3 points will form a triangle. Number of triangles $= {}^{10}C_3 = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 10 \times 3 \times 4 = 120$. Statement I claims the number is 100. Since $120 \neq 100$, Statement I is false. Let's evaluate Statement II: A chord is formed by joining any 2 distinct points on the circle. Number of chords $= {}^{10}C_2 = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 5 \times 9 = 45$. Statement II claims the number is 45. This calculation is correct, so Statement II is true.

Step 4: Final Answer:
Statement I is false and Statement II is true. This corresponds to option (4).