Difference between revisions of "2005 Alabama ARML TST Problems/Problem 10"
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Revision as of 17:31, 1 March 2008
Problem
When Jon Stewart walks up stairs he takes one or two steps at a time. His stepping sequence is not necessarily regular. He might step up one step, then two, then two again, then one, then one, and then two in order to climb up a total of 9 steps. In how many ways can Jon walk up a 14 step stairwell?
Solution
He can either take 14 one-steps, or 12 one-steps and 1 two-step, etc., so we have
$\begin{eqnarray} \frac{14!}{14!}+\frac{13!}{12!\cdot 1!}+\frac{12!}{10!\cdot 2!}+\frac{11!}{8!\cdot 3!}+\frac{10!}{6!\cdot 4!}+\frac{9!}{4!\cdot 5!}+\frac{8!}{2!\cdot 6!}+\frac{7!}{7!}\\ =1+13+66+165+210+126+28+1=14+231+336+29=245+365=\boxed{610} \end{eqnarray}$ (Error compiling LaTeX. Unknown error_msg)
See also
| 2005 Alabama ARML TST (Problems) | ||
| Preceded by: Problem 9 |
Followed by: Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
