Correct : c

This is one of those questions where the difference between the two languages looks very small on the surface, but it completely changes their computational complexity. Let''s break both down carefully.
L1 = {a^mb^mc^m+n | m, n ≥ 1}
In L1, the number of a''s must equal the number of b''s, and the number of c''s must equal m+n. So the count of c''s depends on both m (which is also tied to the count of b''s) and the free variable n. This creates a situation where the automaton must simultaneously track three dependent values — count of a''s, count of b''s (both must be equal to m), and the total count of c''s (which must be m+n).
A pushdown automaton only has a single stack, and it can''t juggle three interdependent counts at the same time. Once it uses the stack to match a''s with b''s, it has lost the information needed to correctly verify the count of c''s. The CFL pumping lemma formally proves L1 is not context-free — any attempt to pump a string in L1 breaks either the a=b equality or the c=m+n condition.
L2 = {a^mbⁿc^m+n | m, n ≥ 1}
Now look at L2. Here, m and n are completely independent of each other — there''s no constraint linking the count of a''s to the count of b''s. The only requirement is that the number of c''s equals the total count of a''s and b''s combined.
This is something a PDA handles very naturally. The PDA simply pushes every a and every b onto the stack as it reads them, and then pops one symbol for each c it encounters. If the stack is empty exactly when all c''s are consumed, the string is accepted. Only one stack operation is needed throughout, making L2 a context-free language.
A simple context-free grammar for L2 would be:
S → AC, A → aAb | ab, C → cC | c (with appropriate adjustments to ensure counts align)
The key takeaway here — whenever you see a language that requires matching counts across three separate sections all tied to the same variable, it''s almost certainly not context-free. But when the counts in one section are a simple sum of two independent variables, a PDA can usually handle it with ease.