7. [30] Determine the number of juggling sequences of length $n$ with exactly 1 ball.
Answer: $\quad 2^{n}-1$. Solution: With 1 ball, we simply need to decide at times should the ball 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 $2^{n}-1$.