Number of compositions

Statement

Let $n$ be a positive integer. Then the number of compositions of $n$ is $2^{n-1}$.

Remarks

Proof: write $n$ as a row of $n$ ones. A composition is a choice of where to cut among the $n - 1$ gaps between them. The number of compositions of $n$ into exactly $k$ parts is $\binom{n-1}{k-1}$. For $n = 0$ there is one composition, the empty one.

Explanations

L1 human

Put $n$ dots in a row. Between two neighbours you either draw a bar or you do not. There are $n - 1$ gaps with $2$ choices each, so there are $2^{n-1}$ compositions.