请用美国人喜爱的幽默风趣的风格制作附近这个题目的讲解视频---**Question Stem:**
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.)
小兔子 Buzz 在台阶上跳上跳下，每次跳一个台阶。 Buzz 从地面开始跳跃，连续跳跃 6 次。 请问 Buzz 有几种方式，最后可以跳回地面？
(例如其中一种方式为上-上-下-下-上-下。)
**Options:**
(A) 4
(B) 5
(C) 6
(D) 8
(E) 12
**Image Description:**
* **Type:** Simple line drawing illustrating the scenario.
* **Main Elements:**
* A cartoon outline of a bunny rabbit is positioned on the left side, facing right.
* A simple line drawing of a staircase with 5 steps is positioned to the right of the bunny. The stairs ascend from left to right. The lowest step appears to be at the same level as the ground where the bunny is depicted.
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答案文本
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Hey there, math adventurers! Meet Buzz Bunny, the most athletic rabbit in the neighborhood! Buzz has discovered this awesome staircase and he's got a challenge for us. He wants to start on the ground, make exactly 6 hops - either up or down one step at a time - and somehow magically end up right back on the ground where he started. Sounds tricky, right? Let's see how many different ways our furry friend can pull off this incredible hopping sequence!
Now here's where it gets really cool! Let's assign some numbers to make this easier. Every time Buzz hops UP, we'll call that plus one. Every time he hops DOWN, we'll call that minus one. Since he makes 6 total hops, if u is the number of up hops and d is the number of down hops, then u plus d equals 6. But here's the kicker - since he ends up back on the ground where he started, all those ups and downs must cancel out perfectly! That means u minus d equals zero, so u equals d. Solving these equations, we get u equals d equals 3. So Buzz needs exactly 3 up hops and 3 down hops!
Plot twist! This problem is actually connected to something super famous in mathematics called Dyck Paths! A Dyck path is like Buzz's journey, but with one crucial rule - he can never go below the ground level. Think of it as a path that starts at zero, uses steps of plus one and minus one, ends at zero, but never dips into negative territory. This constraint makes the problem much more interesting! The number of such paths is given by Catalan numbers. For our case with 6 steps, we use the third Catalan number: C sub 3 equals one-fourth times 6 choose 3, which equals one-fourth times 20, giving us exactly 5 different valid paths!
And here they are - all 5 valid sequences that Buzz can use! Let's watch each one in action. First, up-up-up-down-down-down - Buzz climbs to step 3, then comes all the way back down. Second, up-up-down-up-down-down - he goes up twice, down once, up again, then down twice. Third, up-up-down-down-up-down - similar pattern but different timing. Fourth, up-down-up-up-down-down - he goes up, down, then up twice, then down twice. And finally, up-down-up-down-up-down - the most balanced sequence, alternating most of the way! Each sequence uses exactly 3 ups and 3 downs, never goes below ground level, and brings Buzz safely back home. So the answer is B - 5 different ways!
Hey there, math adventurers! Meet Buzz Bunny, our energetic hopping hero! Buzz has a mathematical challenge that's hopping mad fun. He wants to make exactly 6 hops on these stairs, starting and ending on the ground. Each hop goes either up one step or down one step. But here's the catch - Buzz can never hop below ground level because, well, that would be underground tunneling, not hopping! So the big question is: in how many different ways can Buzz complete this 6-hop adventure and return safely to where he started? Let's hop right into this problem!
Now let's think like mathematicians and break down exactly what Buzz's challenge means. First, since he starts at ground level and must return to ground level after exactly 6 hops, we need the same number of up hops as down hops. That means 3 up hops and 3 down hops - perfectly balanced! But here's the tricky part: Buzz can never dip below ground level during his journey. This constraint is what makes this problem special and much more interesting than just counting all possible arrangements of 3 ups and 3 downs.
Let's visualize what these paths actually look like! I'll show you a valid path: Up-Up-Down-Up-Down-Down. Watch how Buzz's height changes: he starts at 0, goes to 1, then 2, back to 1, up to 2, down to 1, and finally back to 0. Notice that he never dips below the ground line! This is called a Dyck path in mathematics. Now, if Buzz tried a sequence like Up-Down-Down-Up-Up-Up, he would hit level negative 1 after the third hop, which means he'd be underground - definitely not allowed! The key insight is that at every step along the way, the number of up hops must be greater than or equal to the number of down hops.
Now for the mathematical magic! This problem is actually asking for the 3rd Catalan number. Catalan numbers count the number of different ways to correctly match pairs of parentheses, and our up-down hop problem is mathematically equivalent! The formula for the nth Catalan number is C_n equals 1 over n plus 1, times 2n choose n. For our case with n equals 3, we get C_3 equals 1 over 4, times 6 choose 3, which equals 1 over 4 times 20, giving us exactly 5. And here are all five valid hopping sequences: 1. Up-Up-Up-Down-Down-Down, 2. Up-Up-Down-Up-Down-Down, 3. Up-Up-Down-Down-Up-Down, 4. Up-Down-Up-Down-Up-Down, and 5. Up-Up-Down-Down-Down-Up. Each one keeps Buzz safely above ground level throughout his entire journey!
And there you have it, math adventurers! Buzz Bunny has successfully figured out his hopping challenge, and so have we! The answer is B - there are exactly 5 different ways for Buzz to make his 6-hop journey and return safely to the ground. We discovered this through the beautiful mathematics of Dyck paths and Catalan numbers. Remember, this problem required exactly 3 up hops and 3 down hops, with the crucial constraint that Buzz could never go below ground level. From simple counting to advanced combinatorics, we've seen how one little bunny's hopping adventure connects to some really deep mathematical concepts. Thanks for joining Buzz and me on this mathematical adventure - keep exploring, keep questioning, and keep having fun with math!