Miscellaneous

Applying the Binomial Theorem Together

The exercises above practised the binomial theorem, Pascal's triangle, and the special-case expansions one idea at a time. This Miscellaneous Exercise mixes them: some questions ask you to expand a trinomial by first grouping it into a binomial, others use the $(a+b)^n \pm (a-b)^n$ trick to isolate only the even- or odd-placed terms, and a few use the expansion to prove a divisibility result, exactly like Example 4's $6^n - 5n$ proof. Treat each question as a chance to decide, for yourself, which of the chapter's tool …