Question

This question has two parts. First, answer Part A. Then, answer Part B.

Part A
FRIENDSHIPS A small company has 16 employees. The owner of the company became concerned that the employees did not kn
other very well. He decided to make a picture of the friendships in the company. He placed 16 points on a sheet of paper in such a
that no 3 were collinear. Each point represented a different employee. He then asked each employee who their friends were and
connected two points with a line segment if they represented friends.
a. What is the maximum number of line segments that can be drawn between pairs among the 16 points?
Part B
b. When the owner finished the picture, he found that his company was split into two groups, one with 10 people and the other wi
people within a group were all friends, but nobody from one group was a friend of anybody from the other group. How many line
segments were there?

Answer 1

In Part A, the maximum number of line segments that can be drawn between pairs among the 16 points is 120.

In Part B, the number of line segments between the two groups is 60.

Step-by-step explanation:

Part A: In the given scenario, there are 16 employees in the small company. In order to find the maximum number of line segments that can be drawn between pairs among the 16 points, we need to find the number of ways we can choose 2 points from 16.

This can be calculated using the combination formula, which is nC2 = n! / (2!(n-2)!).

Substituting the values, we get 16C2 = 16! / (2!(16-2)!) = 16 x 15 / (2 x 1) = 120.

Part B: In part B, we are given that the company is split into two groups, one with 10 people and the other with 6 people.

Within each group, everyone is friends, but there are no friendships between the two groups.

In order to find the number of line segments between the points representing the two groups, we need to find the number of ways we can choose 1 point from the first group and 1 point from the second group, which is 10 x 6 = 60 line segments.