There are 12 points in a plane such that 3,4,5 of them are...

There are 12 points in a plane such that $3,4,5$ of them are respectively on the sides $\mathrm{AB}, \mathrm{BC}, \mathrm{CA}$ of the triangle $\mathrm{ABC}$. (i) How many triangles can be formed by joining these 12 points? (ii) What is the probability that 3 points selected at random from these points form a triangle?

There are 12 points in a plane such that $3,4,5$ of them are respectively on the sides $\mathrm{AB}, \mathrm{BC}, \mathrm{CA}$ of the triangle $\mathrm{ABC}$.
(i) How many triangles can be formed by joining these 12 points?
(ii) What is the probability that 3 points selected at random from these points form a triangle?

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Key Concepts

Combination Formula

The concept of combinations involves selecting a subset of items from a larger set without regard to the order of selection. In this problem, combinations are used to count the total number of ways to select 3 points out of 12, which is key to enumerating possible triangles.

Degenerate Cases

Degenerate cases refer to configurations that do not meet the criteria for forming the desired structure—in this case, triangles. When three points are collinear, they form a line instead of a triangle. It is crucial to exclude these cases from the total combinations to count only valid triangles.

Collinearity

Collinearity is the property where points lie on the same straight line. Since the points on each side of the triangle are collinear, any three points chosen from the same side do not form a triangle. Identifying collinear sets is essential for determining which combinations do not yield a triangle.

Probability Calculation

Probability in combinatorial problems is computed as the ratio of favorable outcomes to the total number of outcomes. In this context, after determining the number of valid triangles (total combinations minus degenerate cases), the probability that three randomly selected points form a triangle is the fraction of valid triangles over the total combinations.

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