How many point-to-point WANs are needed to connect n LANs if each LAN should be able to directly communicate with any other LAN?
The correct answer and explanation is:
To connect n LANs with point-to-point WANs such that each LAN can directly communicate with every other LAN, we need to create a fully connected (mesh) network.
🔢 Correct Answer:
n(n−1)2\frac{n(n – 1)}{2}
✅ Explanation (300 Words):
In a fully connected network using point-to-point Wide Area Networks (WANs), each LAN must have a direct connection to every other LAN. A point-to-point connection means that there is a dedicated line (or link) between exactly two nodes—in this case, between two LANs.
To find out how many such connections are needed, we use a fundamental concept from combinatorics:
The number of unique pairs that can be formed from n elements is given by:(n2)=n(n−1)2\binom{n}{2} = \frac{n(n – 1)}{2}
Each pair of LANs represents one required point-to-point WAN connection.
💡 Example:
Suppose you have 4 LANs: LAN1, LAN2, LAN3, and LAN4.
Using the formula: 4(4−1)2=4×32=6\frac{4(4 – 1)}{2} = \frac{4 \times 3}{2} = 6
So you need 6 point-to-point WANs to ensure direct communication between all LANs.
🔁 Why Not Use Fewer?
Using fewer WAN connections would mean some LANs have to communicate indirectly (via another LAN), which violates the condition that each LAN must communicate directly with every other.
This setup is expensive and complex in large networks, which is why technologies like switches, routers, or central hubs (like in star topology) are used to simplify connections.
✅ Final Note:
For n LANs, the number of point-to-point WANs needed for full direct communication is: n(n−1)2\boxed{\frac{n(n – 1)}{2}}