2024 AMC 8 Problems/Problem 13

Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of $6$ hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.)

$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 12$

Solution 1

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

UUDDUD

UDUDUD

UUUDDD

UDUUDD

UUDUDD

Counting all the paths listed above gets you 5 or B.

-ALWAYSRIGHT11

Solution 2

Any combination can be written as some re-arrangement of $UUUDDD$ . Clearly we must end going down, and start going up, so we need the number of ways to insert 2 $U$ 's and 2 $D$ 's into $U\, \_ \, \_ \, \_ \, \_ \, D$ . There are ${4\choose 2}=6$ ways, but we have to remove the case $UDDUUD$ , giving us $\boxed{\textbf{(B)}\ 5}$ .

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

~Sahan Wijetunga

Solution 3

These numbers are clearly the Catalan numbers. Since we have 6 steps, we need the third Catalan number, which is $\boxed{\textbf{(B)}\ 5}$ . ~andliu766

Video Solution 1 (easy to digest) by Power Solve

https://youtu.be/X5Xk0wYXypk

Video Solution 2 by Math-X (First fully understand the problem!!!)

https://www.youtube.com/watch?v=Td6Z68YCuQw

~Math-X

Video Solution 3 by OmegaLearn.org

https://youtu.be/dM1wvr7mPQs

Video Solution by CosineMethod [🔥Fast and Easy🔥]

https://www.youtube.com/watch?v=-kCN6R9U944

2024 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

Page

Toolbox

Search