## Challenge 348: Pick a Point

Part (a)

I mark any two points on each side of an equilateral triangle. I pick at random 3 of the six points I have marked.

(i) What is the probability that I pick one from each side of the triangle?

I mark one more point on each side of the triangle and now pick 3 of the nine points at random.

(ii) What is the probability that I pick one from each side of the triangle now?

Now I mark n points on each side of the triangle and again pick 3 of these points at random.

(iii) What is the probability that I pick one from each side of the triangle?

Part (b)

If I don’t mark any points on the triangle but instead just pick 3 points at random from anywhere on its perimeter, what is the probability that I choose three points one on each side of the triangle?

How is this question related to (a) ?