a First, plot one point for each computer and then connect all of the points to each other.
B
b To create the table, count the number of edges of each diagram that you drew on Part A.
C
c Remember that each computer is connected to every other computer and a computer cannot be connected to itself.
D
d Start by looking for a pattern for the number of connections. You can get some help form Part C.
A
aDiagrams:
2 Computers
3 Computers
4 Computers
6 Computers
B
bTable:
Number of Computers
Number of Connections
2
1
3
3
4
6
5
10
6
15
C
c n-1
D
d n(n-1)/2
Practice makes perfect
a We have been given a diagram that illustrates the network if Diego has 5 computers.
We will draw diagrams for the networks if Diego has 2, 3, 4, or 6 computers. To do that, we will first plot one point for each computer and then connect all of the points to each other.
2 Computers
3 Computers
4 Computers
6 Computers
b Using the diagrams that we drew, let's create a table with the number of computers and the number of connections. To do that we will count the number of edges of each diagram.
Number of Computers
Number of Connections
2
1
3
3
4
6
5
10
6
15
c In this part, we will write an expression for the number of computers to which each of the computers connected if there are n computers. We know that each computer is connected to every other computer and a computer cannot be connected itself. Thus, the number of computers with connection is as follows.
n-1
d This time, we will write an expression for the number of connections if there are n computers. Looking at the previous part, we can write a pattern for the number of connections.
The first computer will be connected to the other n-1 computers.
The second computer will be connected to the other n-2 computers since it has already been connected to the first computer.
Similarly, the third computer will be connected to the other n-3 computers and so on.
If we add these connections together, we can find the total number of connections.
(n-1)+(n-2)+(n-3)+...+1
Since the terms of the addition are consecutive numbers, we can use the formula for summation of consecutive numbers and write the final form of our expression.
n(n-1)/2
