For 1000 participants, how many keys are needed in symmetric vs. asymmetric systems?

Symmetric needs 499,500 keys; asymmetric needs only 1000 key pairs.

Symmetric:

$$\frac{n(n-1)}{2} = \frac{1000 \times 999}{2} = 499{,}500 \text{ keys}$$

Asymmetric:

$$n = 1{,}000 \text{ key pairs}$$

One pair per participant is enough, because a message to Bob is encrypted with Bob's public key and a message to Alice with Alice's. The two directions A→B and B→A therefore use different key pairs automatically. A design in which both directions share one key pair must never occur.

If keys are used per direction or per purpose, both counts scale up: directional symmetric keys give $n(n-1) = 999{,}000$, and $t$ separate keys per pair (e.g. one for encryption, one for the MAC) give $t \cdot n(n-1)$. On the asymmetric side, $t$ purposes mean $t \cdot n$ key pairs, e.g. 2,000 for separate encryption and signature pairs.

The scaling problem visualized:

The quadratic growth of symmetric key management is unsustainable for large networks. This is precisely why public key infrastructure (PKI) was invented — it makes large-scale secure communication manageable.