On the definition of degree of closed subschemes

$$\underline {Background}$$:We know that,for a projective variety $$X \subset\mathbb{P}^{n}=(\mathbb{K}^{n+1}-{0})/\sim$$

we define , degree($$X$$)=$$(r!)$$.(leading coefficient of the hilbert polynomial of $$X$$)

$$\underline {Question (1)}$$:What is the definition of degree of a closed subscheme $$X$$ of $$Proj(K[x_0,....,x_n])$$?

Can we define the same thing for closed subscheme?

$$\underline {Question (2)}$$:what can be said about degree of $$0$$-dimensional subcsheme?

Since there are only a finitely many points in a $$0$$ dimensional subscheme ,can we say that in this case degree is same as cardinality?

Finally is there any reference where they talk explicitly about the definition of degree of a closed subscheme in $$Proj(K[x_0,....,x_n])$$(maybe with some example)

Any help from anyone is welcome.