Step 1: Understanding the Question:
The question asks for the standard chemical formulas of the binary compounds produced when lanthanoids (Ln) react with elemental nitrogen and halogens (X).
Step 2: Detailed Explanation:
Lanthanoids predominantly exhibit a stable $+3$ oxidation state in their compounds due to the high stability achieved after losing three valence electrons.
When a lanthanoid reacts with nitrogen (N), which forms a nitride ion ($\text{N}^{3-}$), the $+3$ charge of the metal cation balanced with the $-3$ charge of the anion results in a simple $1:1$ chemical formula:
$$ \text{Ln}^{3+} + \text{N}^{3-} \rightarrow \text{LnN} $$
When reacting with halogens ($\text{X} = \text{F, Cl, Br, I}$), halogens form monovalent halide ions ($\text{X}^-$). To balance the $+3$ charge of a single lanthanoid ion, three halide ions are structurally required:
$$ \text{Ln}^{3+} + 3\text{X}^- \rightarrow \text{LnX}_3 $$
Thus, the respective formulas for the nitride and halide are LnN and $\text{LnX}_3$.
Step 3: Final Answer:
The correct chemical formulas are LnN and $\text{LnX}_3$, which corresponds directly to option (A).