2024 AMC 8 Problems/Problem 13: Difference between revisions
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(For example, one sequence of hops is up-up-down-down-up-down.) | (For example, one sequence of hops is up-up-down-down-up-down.) | ||
<math>\textbf{(A)}\ | <math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 12</math> | ||
==Solution 1== | ==Solution 1== | ||
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-ALWAYSRIGHT11 | -ALWAYSRIGHT11 | ||
==Solution 2== | |||
These numbers are clearly the Catalan numbers. Since we have 6 steps, we need the third Catalan number, which is <math>\boxed{\textbf{(B)}\ 5}</math>. | |||
~andliu766 | |||
==Video Solution 1 (easy to digest) by Power Solve== | ==Video Solution 1 (easy to digest) by Power Solve== | ||
Revision as of 17:06, 26 January 2024
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 
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:
UUDDUD
UDUDUD
UUUDDD
UDUUDD
UUDUDD
Counting all the paths listed above gets you 5 or B.
-ALWAYSRIGHT11
Solution 2
These numbers are clearly the Catalan numbers. Since we have 6 steps, we need the third Catalan number, which is 
.
~andliu766
Video Solution 1 (easy to digest) by Power Solve
Video Solution 2 by Math-X (First fully understand the problem!!!)
https://www.youtube.com/watch?v=Td6Z68YCuQw
~Math-X
