Compositions via binomial coefficients

Stars and bars. Represent $n$ as a sum of $k$ positive parts by placing $k-1$ dividers between $n$ stars in a row of $n$ stars, giving $\binom{n-1}{k-1}$ ways.

Conclusion. Number of compositions of $n$ into $k$ positive parts is $\binom{n-1}{k-1}$.

Total. $\sum_{k=1}^{n}\binom{n-1}{k-1}=2^{n-1}$ compositions of $n$.