Keys required for cryptography I am reading an article on cryptography at following location:
http://www.entrust.com/resources/pdf/cryptointro.pdf

Historically, encryption systems used what is known as symmetric cryptography. Symmetric cryptography uses the same key for both encryption and decryption. Using symmetric cryptography, it is safe to send encrypted messages without fear of interception (because an interceptor is unlikely to be able to decipher the message); however, there always remains the difficult problem of how to securely transfer the key to the recipients of a message so that they can decrypt the message.
With symmetric cryptography, as the number of users increases on a network, the number of keys required to provide secure communications among those users increases rapidly. For example, a network of 100 users would require almost 5000 keys if it used only symmetric cryptography. Doubling such a network to 200 users increases the number of keys to almost
20,000. Thus, when only using symmetric cryptography, key management quickly becomes unwieldy even for relatively small-scale networks.

How did the author conclude 5000 keys for 100 users and 20,000 keys for 200 users?
The system can be viewed as a complete graph, in which each vertex is a user and each edge represents a different symmetric key.

Thus for 100 users (vertices), you need (100*99)/2 keys (edges). This is roughly 5,000.

To help visualise, consider the image below from Wikipedia. It demonstrates a complete graph with seven users. Hence 21 seperate keys would be required.