Let $C$ be a projective curve (not necessarily reduced or irreducible). Let $f:C \hookrightarrow \mathbb{P}^n$ be a closed immersion. Suppose that the degree of $f(C)$ is equal to $e$. How many ways are there to embed $C$ in $\mathbb{P}^{n+1}$ such that the degree of the image is equal to $e$?