Question

A rod is divided into n segments randomly. What is the probability that the segments can form a polygon?

A) 1/n
B) (n - 2)/n
C) (n - 1)/n
D) 1 - 1/n

Answer 1

To form a polygon, the sum of the lengths of any two sides of the polygon must be greater than the length of the remaining side. The probability that the segments can form a polygon is (n - 1)/n.

Step-by-step explanation:

To form a polygon, the sum of the lengths of any two sides of the polygon must be greater than the length of the remaining side. Let's consider the segment lengths as a1, a2, ..., an. For the segments to form a polygon, the sum of the lengths of any two segments must be greater than the length of the remaining segments. Mathematically, this can be represented as:

a1 + a2 > a3, a2 + a3 > a4, ..., an-2 + an-1 > an, an-1 + an > a1

When n segments are randomly divided, there are (n-1)! possible divisions since the first segment can be placed at any position. Each division corresponds to a unique arrangement of segment lengths. Out of these, only one arrangement satisfies the above conditions. Therefore, the probability that the segments can form a polygon is 1/(n-1)!, which simplifies to (n-1)/n. Hence, the answer is option C, (n - 1)/n.