Let $AB, CD, EF, GH,$ and $JK$ be five diameters of a circle with center at $O$ . In how many ways can three points be chosen out of $A, B, C, D, E, F, G, H, J, K,$ and $O$ so as to form a triangle?

Entered answer:

✅ Correct Answer: 160

We have 11 points total: 10 points on the circle (A, B, C, D, E, F, G, H, J, K) plus the center O.

The key insight: Three points form a triangle if and only if they are not collinear (don't lie on the same straight line).

Let's break this down based on whether the center O is selected or not.

If we don't choose O, we're selecting 3 points from the 10 points on the circle.

Total combinations: ${{}}^{10}C_{3} = \dfrac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$

All these are valid triangles because any three points on a circle always form a triangle. No three points on a circle can be collinear.

So we get 120 triangles when O is not selected.

Now we need to choose 2 more points from the remaining 10 points on the circle.

Important constraint: The two points we choose cannot be diametrically opposite because then all three points (the two chosen points + O) would be collinear.

Total ways to choose 2 points from 10: ${{}}^{10}C_{2} = 45$

Invalid ways (diametrically opposite pairs): 5 pairs (AB, CD, EF, GH, JK)

Valid ways: $45 - 5 = 40$

Alternative approach: Pick any point (say A) and pair it with 8 other points (all except B, its diametrically opposite). We have 10 points, each can be paired with 8 others: $10 \times 8 = 80$ ordered pairs. Since we want unordered pairs: $\dfrac{10 \times 8}{2} = 40$

So we get 40 triangles when O is selected.

Total number of triangles = 120 + 40 = 160

Answer: 160 ways

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Let AB,CD,EF,GH,AB, CD, EF, GH,AB,CD,EF,GH, and JKJKJK be five diameters of a circle with center at OOO. In how many ways can three points be chosen out of A,B,C,D,E,F,G,H,J,K,A, B, C, D, E, F, G, H, J, K,A,B,C,D,E,F,G,H,J,K, and OOO so as to form a triangle?

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CAT Past Year Questions (Topic-Wise)

Let $AB, CD, EF, GH,$ and $JK$ be five diameters of a circle with center at $O$ . In how many ways can three points be chosen out of $A, B, C, D, E, F, G, H, J, K,$ and $O$ so as to form a triangle?