The nodes do not modify the birational invariants of a surface. So if we blow-up the $k$ nodes of $X$ we obtain a smooth K3 surface $S$, whose topological Euler number is $24$. Coming back to $X$, we substitute each $(-2)$-curve (which is topologically a sphere, so has Euler number $2$) with a point. So the Euler number of $X$ is $24-k$.

The nodes do not modify the birational invariants of a surface. So if we blow-up the $k$ nodes of $X$ we obtain a smooth K3 surface $S$, containing $k$ $(-2)$-curves, whose topological Euler number is $24$. Coming back to $X$, we substitute each $(-2)$-curve (which is topologically a sphere, so has Euler number $2$) with a point. So the Euler number of $X$ is $24-k$.