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Asked: 2024-09-25T01:31:42+05:30

You are faced with a staircase that has a certain number of steps, and you have the ability to climb either one step or two steps at a time. Your challenge is to determine the total number of distinct ways you can reach the top of the staircase. Given an integer representing the total number of steps, calculate all the possible combinations of steps that can lead you to the summit. How many unique paths can you take to conquer the staircase?

Imagine you’ve found yourself standing in front of a tall staircase. It’s not just any staircase; this one has a total of, let’s say, 5 steps leading to the top. You’re feeling pretty adventurous today and have decided to tackle the climb. But here’s the catch—you can either take one step at a time or leap two steps at once whenever you feel bold enough!

As you look at the staircase, you can’t help but wonder: how many different ways can you reach the very top?

Maybe you start by taking one step at a time, which feels safe and steady, giving you the sense that you’re in control. Or perhaps you think about taking some two-step leaps to speed things up. It’s kind of exciting to think about all the combinations.

To make it even more interesting, you can take a little detour in your thought process: what if the staircase had 6 steps instead? Would that change the number of ways you could reach the top? Or what if it had 10 steps?

Picture this. You could go up step 1, then step 2, and then decide to jump to step 4, or perhaps you’ll take the two steps consecutively at the beginning and then go for smaller steps afterward. The variety of combinations feels like a fun puzzle just waiting to be solved!

So here’s the challenge: Are you up for it? How would you go about calculating all those unique paths? And more importantly, can you figure out how many distinct ways there are to conquer a staircase with, let’s just throw a number out there, 5 steps?

Thinking about this can really get the brain churning! What strategies would you use? Would you draw it out, make some notes, or just count it all up in your head? I’d love to hear how you would tackle this little stair-climbing scenario! What’s your answer?

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    2 Answers



    1. Staircase Climbing Challenge

      How Many Ways to Climb a Staircase?

      So, I’m standing here in front of this staircase with 5 steps. I can either take it easy and go one step at a time, or I could jump a couple of steps to make things quicker. It’s kind of exciting to think about all the different ways to get to the top!

      I think the possibilities could look something like this:

      • Taking 5 single steps: 1, 2, 3, 4, 5
      • Maybe I take a mix like: 1, 2, and then leap to 4. Hmm, that’s 3 moves!
      • Or I could start with a leap: 1, 2, then leap to 4, and finally take 5. That’s pretty cool!

      It feels like I can keep doing this in my head, taking different paths. But to make it more fun, if I had 6 steps, I could easily add another leap or two. And then if I had 10 steps? Oh man, that could get wild!

      Your mind kind of runs in circles figuring out how many ways to reach the last step. After daydreaming a bit, I think it can be calculated? Like, maybe using some math or a simple program?

      Counting the Ways

      Looking it up a bit, it sounds like it might be related to Fibonacci numbers? For 5 steps, I think the calculations can go like this:

      • 1 step to Step 4 + 1 single step to Step 5
      • 2 steps to Step 3 + 2 steps to Step 5

      So if we break it down, it’s kinda like a sequence.

      Concrete Example

      In simple terms:

      • For 1 step – 1 way
      • For 2 steps – 2 ways (1+1 or 2)
      • For 3 steps – 3 ways (1+1+1, 1+2, 2+1)
      • For 4 steps – 5 ways
      • For 5 steps – 8 ways!

      So, there you go! I think it’s 8 ways to get to the top! I really like how this little challenge stretches my brain and makes me think a bit outside the box.

      Can’t wait to tackle the steps on a bigger staircase!


    2. To determine the number of distinct ways to climb a staircase of a given number of steps, we can use a recursive approach or dynamic programming. The fundamental principle is that from any given step “n”, you can arrive there either from step “n-1” (taking a single step) or from step “n-2” (taking a leap of two steps). Hence, the total number of unique ways to reach the top of a staircase with “n” steps can be expressed as the sum of the ways to reach the two preceding steps: ways(n) = ways(n-1) + ways(n-2). For base cases, we have ways(1) = 1 (only one way to step on the first step) and ways(2) = 2 (either take two single steps or one leap of two steps). Using this logic, we can derive the number of distinct paths for a staircase with 5 steps.

      For a staircase with 5 steps, the sequence of calculations would be as follows: ways(3) = ways(2) + ways(1) = 2 + 1 = 3, ways(4) = ways(3) + ways(2) = 3 + 2 = 5, and finally, ways(5) = ways(4) + ways(3) = 5 + 3 = 8. Therefore, there are 8 different ways to reach the top of a 5-step staircase. Extending this reasoning to 6 steps, we’d find ways(6) = ways(5) + ways(4) = 8 + 5 = 13, leading to 13 unique paths. For 10 steps, the calculations would yield ways(10) = 89 unique paths to reach the top. This combinatorial approach provides an elegant solution to the problem, allowing for a comprehensive understanding of how such scenarios can generate varying paths based on the number of steps.