Question

18 Points are indicated on the perimeter of a triangle ABC as shown below. If three points are chosen then probability that it will form a triangle is

• A. \( \frac{1}{2} \)
• B. \( \frac{355}{408} \)
• C. \( \frac{331}{816} \)
• D. \( \frac{711}{816} \)

Hint:

When choosing points from a set where collinearity is a concern, first calculate the total number of choices and then subtract the number of invalid (collinear) cases.

Answer 1

The Correct Option is D

We are given 18 points on the perimeter of a triangle AB(C) We need to find the probability that three points chosen from these 18 will form a triangle. Step 1: Total number of ways to choose 3 points The total number of ways to choose 3 points from 18 is given by the combination formula: \[ \binom{18}{3} = \frac{18 \times 17 \times 16}{3 \times 2 \times 1} = 816 \] Step 2: Invalid choices (collinear points) To form a triangle, the 3 points must not be collinear. The points are located on the perimeter of the triangle, so collinear points will lie on the sides of the triangle. For each side, we can choose 3 points in the following way: - There are 6 points on each side, and the number of ways to choose 3 collinear points from 6 is: \[ \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \] Since there are 3 sides to the triangle, the total number of ways to choose 3 collinear points (which will not form a triangle) is: \[ 3 \times 20 = 60 \] Step 3: Valid choices (forming a triangle) The number of valid ways to choose 3 points that form a triangle is the total number of ways to choose 3 points minus the number of collinear choices: \[ 816 - 60 = 756 \] Step 4: Probability The probability that the 3 points chosen will form a triangle is the ratio of valid choices to total choices: \[ \frac{756}{816} = \frac{711}{816} \] Thus, the correct answer is \( \boxed{\frac{711}{816}} \).