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Let $n$ be a positive integer. Using induction, prove that there are $2^{(nāˆ’1)}$ tuples $(m_1, \ldots , m_k)$ of positive integers with the property that $m_1 + \cdots + m_k = n$.

Alain Remillard
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Ryan
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    Surely you mean $2^{n-1}$ instead? Here is a hint... $3+1+2$ could be written as $\circ \circ \circ \color{red}{+}\circ\color{red}{+}\circ\circ$. If you insist on induction, then consider either taking a tuple adding to $n-1$ and increasing the final value by one, or adding an additional $1$ by itself at the end. ā€“ JMoravitz Oct 10 '19 at 18:32

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Hint: $n = (m_1 + ....+m_k)$ can written $2^{n-1}$ ways,

Then $n+1 = 1+(m_1 + ....+m_k)$ can also be written in $2^{n-1}$ ways.

How many ways can $n+1 = 2+ (m_1 + ..... + m_k)$ be written?

What about $n+ 1 = n + (m_1+..... + m_k)$?

And what about for some $1 < j < n$, then how many ways can $n+1=j + (m_1 + ..... + m_k)$ be written?

fleablood
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