Symmetric key management

A general question I always have in my mind about symmetric key management is how many keys are required for a network with n network nodes. People familiar with cryptography can say right the wayit is ( n * (n-1)/2). I am too old to recall the mathematics behind it . What I can do is to draw a picture to prove that in a 4-node network, there are 6 keys required in order to help me to memorize n * (n-1) /2.



This generally raises the question of the immense difficulty in symmetric key management if the number of network node is large, lets say over 20. Yes, that's why asymmetric key in the form of a private key paired with a public key (PKI) is commonly preferred for secure communications over the Internet.