Competition problem — Combinatorics

Combinatorics Difficulty 4.7 AIME Prove it United States

Problem:
Determine the number of juggling sequences of length $n$ with exactly $1$ ball.

Solution

Solution:
With $1$ ball, we simply need to decide at which times the ball should land in our hand. That is, we need to choose a non-empty subset of $\{0, 1, 2, \ldots, n-1\}$ where the ball lands. It follows that the answer is $2^{n} - 1$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.