2024 AMC 8 Problems/Problem 13

Problem

Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz Bunny start on the ground, make a sequence of hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.)

Solution 1

Looking at the answer choices, you see that you can list them out. Doing this gets you:

Counting all the paths listed above gets you .

~ALWAYSRIGHT11

Solution 2

Any combination can be written as some re-arrangement of . Clearly we must end going down, and start going up, so we need the number of ways to insert 2 's and 2 's into . There are ways, but we have to remove the case , giving us .

- We know there are no more cases since there will be at least one before we have a (from the first ), at least two 'S before two 's (since we removed the one case), and at least three 's before three 's, as we end with the third .

~Sahan Wijetunga

Solution 3

These numbers are clearly the Catalan numbers. Since we have steps, we need the third Catalan number, which is . ~andliu766

Solution 4

The first step must be a U, last step must be a D. After the third step we can get only position 3 or position 1. In the first case there is only one way: UUUDDD. In the second case we have two way to get this position UDU and UUD. Similarly, we have two way return to position 0 (UDD and DUD). Therefore, we have .

Solution 5 (Complementary Counting)

We can find the total cases then deduct the ones that don't work.

Let represent "Up" and represent "Down". We know that in order to land back at the bottom of the stairs, we must have an equal number of 's and 's, therefore six hops means of each.

The number of ways to arrange 's and 's is .

Case : Start with

Case : Start with

Case : Start with

Case : Start with

Case is asking us how many ways there are to arrange 's and 's, which is .

Case is asking us how many ways there are to arrange 's and , which is .

Case is asking us how many ways there are to arrange , which is .

Case is asking us the same thing as Case , giving us .

Therefore, deducting all cases from gives .

~Tacos_are_yummy_1

Video by MathTalks😉

https://youtu.be/qAwRUj2N46c?si=QDUY8ZUVFP29Eg4c

~rc1219

Video Solution 1

https://youtu.be/BaE00H2SHQM?si=GTocuz7rsKFCrPn3&t=2986 ~Math-X

Video Solution 2 by Central Valley Math Circle(Goes through the full thought process)

https://youtu.be/aMmHkyEOmAc ~mr_mathman

Video Solution 3 (A Clever Explanation You’ll Get Instantly)

https://youtu.be/5ZIFnqymdDQ?si=n9jyl0QHXLbaKz3I&t=1363

~hsnacademy

Video Solution 4 by OmegaLearn.org

https://youtu.be/dM1wvr7mPQs

Video Solution 5 (Easy to understand!)

https://www.youtube.com/watch?v=V-xN8Njd_Lc ~NiuniuMaths

Video Solution 6 Interstigation

https://youtu.be/ktzijuZtDas&t=1238

Video Solution 7 Dr. David

https://youtu.be/r3FtOOYEces

Video Solution 8 WhyMath

https://youtu.be/6Bg0Z0jcInw

Video Solution 9 (30-Second Rearrangement + 2 other methods)

https://www.youtube.com/watch?v=WhuMrsxkf70 ~TheMathGeek

Video Solution by Daily Dose of Math (Simple, Certified, and Logical)

https://youtu.be/qJqrFauo3lQ

~Thesmartgreekmathdude

See Also

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.