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Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to Grid Paths
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Let’s start our discussion about grid paths! Can anyone tell me what we mean by paths in a grid?
Are those the different ways we can move from one corner of the grid to the opposite corner?
Exactly! In a rectangular grid, we can move either right or up. If we start at (0,0) and aim to reach (m,n), we have to make 'm' moves to the right and 'n' moves up.
So how do we figure out how many different ways we can do that?
Great question! We calculate this using combinations. The total number of moves you need to make is 'm+n' and then you can choose positions for the 'm' right moves, which gives you \(C(m+n, m)\).
And what does \(C\) stand for?
Good catch! \(C(n,k)\) represents combinations, which is the number of ways to choose 'k' items from 'n' items. In our case, that's how we calculate the paths.
So if we have a 5 column and 10 row grid, how many paths are there?
That would be \(C(15, 5)=3003\). So remember, the formula is key here!
To summarize: We can calculate paths using combinations and learn that in a grid with 'm' columns and 'n' rows, total paths are \(C(m+n, m)\).
Blocked Intersections
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Now, let’s talk about what happens when there are blocked intersections, like (2,4). What do we do then?
Do we just exclude the paths that go through that intersection?
Exactly! We can calculate how many paths go through the blocked intersection and then subtract that from the total. But first, can anyone tell me how to find paths passing through (2,4)?
We would need to find paths from (0,0) to (2,4) and then from (2,4) to (5,10).
Correct! So we compute the two segments: \(C(6,2)\) for the first segment and \(C(9,3)\) for the second.
And then we multiply those two values together to find all paths that pass through that point?
Right! So if we multiply these, we would calculate the invalid paths and to compute the valid paths, we perform: total paths minus these invalid ones. What do we end up with if the total is 3003 and invalid paths are 1260?
That would be 1743 remaining valid paths!
Exactly! Well done! So remember, counting invalid paths helps us filter out valid paths in grids with blocked intersections.
In summary: Blocked intersections require us to calculate invalid paths through them and subtract from the total. We keep adjusting our calculations based on new blocks entering the picture.
Multiple Blocked Intersections
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Now that we’ve tackled single block paths, what happens if we have two blocked intersections? For example, (2,4) and (4,4)?
Wouldn't we just add blocked path counts together?
That's a common mistake! We need to subtract paths through both intersections because we’ll end up counting some paths twice. Can anyone tell me what that principle is called?
I think it’s inclusion-exclusion!
Correct! We include individual blocked paths and exclude the paths traversing both blocks.
How do we find those paths that go through both?
You’d compute the paths from (0,0) to (2,4) then to (4,4) and finally to (5,10). It’s a combination of three segments. Path count through both is then added back after subtraction. What’s our result if we calculate that?
So we learn to adjust our block counts to keep everything intact!
Exactly! Always be cautious of double counts, using inclusion-exclusion helps us keep our math aligned. Well done!
To recap: For multiple blocked intersections, use inclusion-exclusion to accurately calculate and adjust path counts.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section presents the classic problem of calculating the number of unique paths in a grid from the bottom left to the top right corner, exploring combinatorial techniques. Special attention is given to paths that must avoid blocked intersections, and the application of these combinatorial concepts is illustrated through examples and inclusion-exclusion principles.
Detailed
Paths with a Blocked Intersection
In this section, we explore how to compute paths on a rectangular grid from the bottom left corner (0,0) to the top right corner (m,n), only allowing movements to the right and up. With m columns and n rows, the total number of unique paths can be calculated using combinatorial principles based on binomial coefficients.
Key Concepts Covered:
- Basic Path Calculation: The total number of paths from (0,0) to (m,n) is given by the formula \(C(m+n, m)\), where \(C(n, k) = \frac{n!}{k!(n-k)!}\). For a 5-column, 10-row grid, there are 3003 paths.
- Blocked Paths: When an intersection, such as (2,4), is blocked, paths that pass through this point must be excluded from the total count. We categorize valid paths by first calculating invalid paths that traverse through the blocked point.
- Inclusion-Exclusion Principle: In cases where multiple intersections are blocked, we compute the number of paths through each intersection and use the inclusion-exclusion principle to avoid double-counting paths that may pass through both blocked points.
Through these combinatorial methods, students can effectively solve more complex grid path problems while considering constraints such as blocked intersections.
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Audio Book
Dive deep into the subject with an immersive audiobook experience.
Introduction to Paths in a Grid
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So, we have a grid, a rectangular grid, where we start walking at the bottom left corner, and the rule is that we can only go up or right. So, we want to start at the bottom and we want to reach the top right corner. So, we can number the coordinates. So, the bottom right corner we call (0,0). This particular grid has got 5 columns and 10 rows. So, the top right corner is (5, 10). And the question that we ask is how many different ways are going, are that go from (0, 0) to (5, 10).
Detailed Explanation
In this chunk, the problem is introduced. We start at the lower left corner of a grid (0,0) and want to reach the upper right corner (5,10) only moving up or to the right. The task is to find out how many different paths exist to reach this destination considering some constraints.
Examples & Analogies
Think of it like someone who is trying to walk through a city grid, starting from a point (like a park at the bottom left) to reach a tall building at the top right. The only way the person can move is either up a street (like going north) or over a street (like going east). The goal is to count all possible ways to navigate the streets to reach that building.
Understanding Combinations of Moves
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Now, it turns out this problem is actually a very classical problem in combinatorics counting. So, the way to analyze this is to see, that if I want to go from the bottom to the top, then I must, on the, in one-dimensional I must go from 0 to 5. In the other dimensions, I must go to 0 to 10. So, totally I must make 15 steps… So, therefore, all I need to do to determine exactly which path I am taking is to fix the position of the 5 rights moves among the total 15, right. So, among 15 positions I choose 5. This is usually written as 15 choose 5, right.
Detailed Explanation
This chunk dives into the mathematical approach to counting the paths. Since you must move a total of 15 steps (5 rights and 10 ups), we need to fix the positions of the right moves. This is expressed using combinatorial notation (15 choose 5), which calculates how many ways we can choose 5 positions for rights out of 15 total steps.
Examples & Analogies
Imagine writing a program where you have to list your favorite 5 out of 15 movies. The combinations you can create based on your favorite choices represent how you can choose to move right in the grid, while the rest can be understood as the upwards movements. Each unique selection leads to a different 'path' to goshows your preferences.
Dealing with Blocked Intersections
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So, what, for example, if this is not a perfect grid. Supposing we have some intersections, which are blocked. So, in this particular case, if you look at this intersection, which is (2, 4), right, it is 1, 2 and then 1, 2, 3, 4. So, we have put a black mark to indicate that for the moment, you cannot go through it, right. So, any part that goes through (2, 4), should not be counted among the valid path from (0, 0) to (5, 10).
Detailed Explanation
This chunk introduces an important constraint in the problem: blocked intersections. For the example given, the intersection at (2,4) is blocked, meaning any path that crosses this point cannot be counted as a valid path. The challenge is to adjust the total path calculations to exclude those that pass through blocked intersections.
Examples & Analogies
Think of this like a construction site blocking a street in a town. If the original routes from a park to the tall building realistically pass through that construction site, you cannot count those routes as valid, since they are not usable. You must find alternatives that avoid the blocked area altogether.
Calculating Valid Paths with Blocked Points
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So, it turns out that actually this also has a combinatorial solution. So, what you can say is that in order to go from here and if I want to avoid a block intersection... So, I get 6 choose 2. So, I get there are 15 ways to go from (0, 0) to (2, 4). Likewise, if I consider this grid, then this is 3 and this is 6 because it was 10 and 5, I have done 2 and 4 respectively... So, this is total number of paths which are going through (2,4), but I do not want paths going through (2,4).
Detailed Explanation
In this chunk, we learn how to count valid paths despite blocked intersections. We calculate the total paths that go through the blocked intersection and subtract those from the initial total. By doing so, we get the number of valid paths that avoid the intersection altogether.
Examples & Analogies
Continuing the street analogy, if you know that there are a certain number of routes that go through a blocked street, you can simply subtract that number from the total options you had at the start. This way, you can find out exactly how many paths you can still walk without running into construction.
Account for Multiple Blocked Intersections
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So, this imperfection could be more complex. I could have two positions blocked, right... But now what happens is, this yellow path is subtracted twice because it is part of the paths going through (2,4) and part of the paths going through (4,4), right. So, I have accidentally removed it twice from my total, so I have to put it back. You have to now compute those paths, which go through both the intersections and add them back.
Detailed Explanation
When there are more than one blocked intersections, this chunk describes a more advanced counting method called inclusion-exclusion. It involves first subtracting paths that go through either blocked point, but making sure that paths that cross both points are added back in because they were subtracted twice.
Examples & Analogies
Think of filtering through all possible routes in a scenario where two routes are both blocked. If you remove all paths that go through either block but accidentally exclude those that might have crossed both cities, it's like trying to figure out the real number of valid routes when both towns have road closures.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Basic Path Calculation: The total number of paths from (0,0) to (m,n) is given by the formula \(C(m+n, m)\), where \(C(n, k) = \frac{n!}{k!(n-k)!}\). For a 5-column, 10-row grid, there are 3003 paths.
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Blocked Paths: When an intersection, such as (2,4), is blocked, paths that pass through this point must be excluded from the total count. We categorize valid paths by first calculating invalid paths that traverse through the blocked point.
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Inclusion-Exclusion Principle: In cases where multiple intersections are blocked, we compute the number of paths through each intersection and use the inclusion-exclusion principle to avoid double-counting paths that may pass through both blocked points.
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Through these combinatorial methods, students can effectively solve more complex grid path problems while considering constraints such as blocked intersections.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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In a 5 by 10 grid, calculate the total unique paths from (0,0) to (5,10), resulting in 3003 paths.
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For a grid with a blocked intersection at (2,4), calculate the total paths avoiding this intersection by first calculating all paths through (2,4) and subtracting from the total.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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To travel up or move right, choose your steps with delight!
📖 Fascinating Stories
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Imagine a brave traveler journeying from the bottom of a grand city grid to the top, dodging obstacles in their path with wise strategies!
🧠 Other Memory Gems
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Remember 'Right and Up are the paths to choose' — RUP helps us recall our options while navigating grids!
🎯 Super Acronyms
C.A.R.E (Combinations Always Result in Easy paths) reminds us how to choose paths efficiently!
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Grid Path
Definition:
A route taken through a grid moving only in right and upward directions.
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Term: Combination
Definition:
A selection of items from a larger set where the order does not matter, represented mathematically as \(C(n,k)\).
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Term: Blocked Intersection
Definition:
A point in the grid that cannot be passed through when calculating paths.
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Term: InclusionExclusion Principle
Definition:
A combinatorial technique used to count the size of the union of overlapping sets.