7c3trianglesprobability.ts

// 7 points lay equally distanced on a circle. 3 points are chosen at random. What’s the probability that the triangle formed by the 3 points is acute?

const points: [x: number, y: number][] = [];

for (let i = 0; i < 7; i++) {
  const angle = Math.PI * 2 * (i / 7);

  points.push([Math.cos(angle), Math.sin(angle)]);
}

const combos = nck(7, 3);

const triangles = combos.map((indices) => indices.map((index) => points[index - 1]));

let count = 0;

for (const [a, b, c] of triangles) {
  const sides = [
    Math.hypot(a[0] - b[0], a[1] - b[1]),
    Math.hypot(b[0] - c[0], b[1] - c[1]),
    Math.hypot(c[0] - a[0], c[1] - a[1]),
  ];

  const [short1, short2, long] = sides.sort((a, b) => a - b);

  if (short1 ** 2 + short2 ** 2 > long ** 2) count++;
}

// ANSWER: 2 / 5 or 0.4
console.log(count / triangles.length);

function nck(n: number, k: number) {
  const result: number[][] = [];
  const combos: number[] = [];

  const recurse = (start: number) => {
    if (combos.length + (n - start + 1) < k) {
      return;
    }
    recurse(start + 1);
    combos.push(start);
    if (combos.length === k) {
      result.push(combos.slice());
    } else if (combos.length + (n - start + 2) >= k) {
      recurse(start + 1);
    }
    combos.pop();
  };

  recurse(1);
  
  return result;
}