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Nice work. Certainly quicker than the approach I showed previously.

A bit heavy for two marks though.

Alternative method for part b: \begin{align*} \text {RHS} &= 2^m $${n \choose m} \\ &=\Bigg[ $${m \choose 0}+$${m \choose 1}+$${m \choose 2}+...+$${m \choose m}\Bigg] $${n \choose m} \\ &=$${n \choose m}+$${m \choose 1}$${n \choose m}+$${m \choose 2}$${n \choose m}+...+$${n \choose m} \\...