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Difficulty: MediumCombinations

1010 distinct points are marked on the circumference of a circle. How many different triangles can be formed by connecting any 33 of these points as vertices?

  1. A
    4545
  2. 120120Answer
  3. C
    360360
  4. D
    720720

Answer

The number of distinct triangles that can be formed is 120120.
To form a triangle, any 33 points must be chosen from the 1010 available points. Because no three points on a circle are collinear, every choice of 33 points forms a unique triangle. Since the order of choosing vertices does not alter the triangle, we use combinations: (103)=10×9×83×2×1=120\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120.

Step-by-Step Solution

1
Identify the total number of items nn and the subset size rr.
n=10n = 10 points on the circle, and r=3r = 3 points required to form a triangle.
Any set of 33 non-collinear points uniquely determines a triangle. Since all points lie on a circle, no three points are collinear.
2
Determine whether order matters.
Order does not matter because choosing points A,B,CA, B, C produces the same triangle as choosing B,C,AB, C, A.
Selection of vertices for a geometric shape is a combination problem, not a permutation problem.
3
Apply the combination formula (nr)=n!r!(n−r)!\binom{n}{r} = \frac{n!}{r!(n - r)!}.
(103)=10×9×83×2×1=7206=120\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = \frac{720}{6} = 120
Evaluating the combination gives the total number of distinct triangles.

Key Concept

Combinations for Geometric Formations
Estimated Time:1m 30s
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Topics

Combinations

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