A group of 30 people who wish to establish pair-wise secure communications using symmetric-key cryptography. How many KEYS need to be exchanged in total?
For symmetric pairwise, the minimum would be 435 unique keys. (N * (N-1)) / 2 unique connections between nodes.
For asymmetric, it would be 30 keys. Which is one reason asymmetric encryption is so useful.
Let's start off with the fact that there are $n\cdot n$ ordered pairs of (person, person). Now we assume that nobody needs encryption to talk to themselves, so we subtract the $n$ pairs where the person is the same, to get $n\cdot n-n=n(n-1)$. Next, we note that the same secret key can be used to send messages on the route (A,B) as on the route (B,A) (with proper care taken to avoid reusing the same nonce, etc). So we divide by two to get $n(n-1)/2$. Finally, we plug in $n=30$ to get $30\cdot 29/2=15\cdot 29=435$.