 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-3/4i/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The language in (ii) is context-free. The grammar

S \to a S c ∣ B, B \to bb B ∣ ϵ

generates exactly

{a^{n} b^{2 i} c^{n} : n, i \geq 0} .

The recursive uses of the first rule match the numbers of

a

's and

c

's, while

B

independently generates any even number of

b

's.

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