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The currency in Binaryland uses only coin denominations that are powers of 2 (such as 1 cent, 2 cents, 4 cents, 8 cents, and so on).
How many ways can one make 8 cents using these coin denominations?

My attempt:

\(\begin{array}{|c|c|c|c|c|} \hline n & 1\ \text{Cent} & 2\ \text{Cent}& 4\ \text{Cent} & 8\ \text{Cent} \\ \hline 1 & & & & 8 \\ 2 & & & 2 & \\ 3 & & 2 & 1 & \\ 4 & 2 & 1 & 1 & \\ 5 & 4 & 0 & 1 & \\ 6 & & 4 & & \\ 7 & 2 & 3 & & \\ 8 & 4 & 2 & & \\ 9 & 6 & 1 & & \\ 10 & 8 & & & \\ \hline \end{array}\)

Enumerate possible combinations:

{8}

{4, 4}

{4, 2, 2}

{4, 2, 1, 1}

{4, 1, 1, 1, 1}

{2, 2, 2, 2}

{2, 2, 2, 1, 1}

{2, 2, 1, 1, 1, 1}

{2, 1, 1, 1, 1, 1, 1}

{1, 1, 1, 1, 1, 1, 1, 1}

Total 10 ways.