Hopping up ten stairs in an even number of jumps

Question

A rabbit at ground level wants to reach the top of a $10$ -step staircase. Each jump can climb any positive number of steps, but the total number of jumps must be even. How many different jump sequences take it exactly to the top of step $10$ ?

Short answer

$256$ . A jump sequence is determined by which of the $9$ intermediate steps it lands on. An even number of jumps means an odd number of landing steps, and exactly half of the $2^9$ subsets of a $9$ -element set are odd-sized.

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Lessons behind this question

Counting and Combinatorics
Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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