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:

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

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.