Set of unordered $n$-tuples of points of $\mathbb{CP}^1$ can be identified with $\mathbb{CP}^n$?

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This is a followup question to here.

Fix $n \in \mathbb{N}$. What is the easiest way to see that the set of unordered $n$-tuples of points of $\mathbb{CP}^1$ can be identified with $\mathbb{CP}^n$?