Number of keys when using symmetric and asymmetric encryption?
For the symmetric key, you can approach this problem as a complete graph with order 1000. With the vertexes representing people and the edges representing the symmetric keys. Then each vertex would have degree 999 and, applying the Handshaking lemma, the number of edges would be:

$(1000 \times 999)/2 = 499500$

So they would need 499500 symmetric keys to have a secure communication between all of them.

For the asymmetric keys, each one would have 2 keys, so a total of 2000 keys.