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A fully connected network topology is a topology in which there is a direct link between all pairs of nodes. Given a fully connected network with $n$ nodes, the number of direct links as a function of $n$ can be expressed as

1. $\frac{n(n+1)}{2}$
2. $\frac{(n+1)}{2}$
3. $\frac{n}{2}$
4. $\frac{n(n-1)}{2}$

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If there are $n$ nodes then

We can connect $node\ 1$  to $n-1$ nodes using $n-1$ direct links

We can connect $node\ 2$ to $n-1$ nodes using $n-2$ direct links ( not $n-1$ since we have already connected it with $node 1$)

We can connect $node\ 3$ to $n-1$ nodes using $n-3$ direct links ( not $n-1$ since we have already connected it with $node\ 1$ and $node\ 2$)

.....

We can connect $node\ n$  to $n-1$ nodes using $0$ direct links (since it is already connected to every other node)

So total number of links required = $(n-1)+(n-2)+(n-3)+....+1+0= \frac{n(n-1)}{2}$

$\therefore$ Option $4.$ is correct.

by Boss (21.5k points)
4. n(n-1)/2
by (19 points)

a direct link between all pairs of nodes , it clearly indicates that whatever the topology looks like,

it always with a Complete graph with 'n' nodes.

the number of direct links as a function of n , they simply ask the number of edges .

So, the number of edges in a complete graph with 'n' nodes = (n*(n-1))/2.

Hence , OPTION D.

by (317 points)