How to prove Binomial Theorem

Question

I have got through part a) and b) The answer is 365 and 364, respectively , however I'm not able to tackle part c)

Answer 1

\begin{align} A-B&=2^{2n}\binom{2n}0-2^{2n-1}\binom{2n}1+2^{2n-2}\binom{2n}2-2^{2n-3}\binom{2n}3+\cdots-2\binom{2n}{2n-1}+\binom{2n}{2n}\\ &=\sum_{k=0}^{2n} \binom{2n}k(-1)^k2^{2n-k}. \end{align} The binomial theorem states that $$(x+y)^m=\sum_{k=0}^m \binom mkx^{m-k}y^k.$$ Does that look useful?

I think you should also work out $A+B$.