Consider the following languages: $$L_1 = \{a^n b^m c^{n+m} : m, n \ge 1\}$$ $$L_2 = \{a^n b^n c^{2n} : n \ge 1\}$$ Which one of the following is TRUE?

Let us analyze both languages:

1. Language $L_1 = \{a^n b^m c^{n+m} : m, n \ge 1\}$:

   • The number of $c$'s equals the sum of the number of $a$'s and $b$'s ($|c| = |a| + |b|$).

   • A Pushdown Automaton (PDA) can push all incoming $a$'s and $b$'s onto the stack and pop one symbol from the stack for each matching $c$.

   • A Context-Free Grammar (CFG) generating $L_1$ can be written as:

     $$S \rightarrow aSc \mid aAc$$

     $$A \rightarrow bAc \mid bc$$

   • Therefore, $L_1$ is a context-free language.

2. Language $L_2 = \{a^n b^n c^{2n} : n \ge 1\}$:

   • A single stack PDA cannot simultaneously verify that the number of $a$'s equals the number of $b$'s and that the number of $c$'s equals $2n$.

   • Using the Pumping Lemma for CFLs, one can show that pumping will disrupt either the count equality of $a$'s and $b$'s or the proportion of $c$'s.

   • Therefore, $L_2$ is not context-free.

Thus, $L_1$ is context-free while $L_2$ is not context-free.

Therefore, the correct option is (B).