1. Grid Paths

So, continuing our discussion about efficiently computing recursive functions, let us look at the problem of computing grid paths. 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).

In this section, we are introduced to the concept of grid paths, which involves moving within a rectangular grid starting from the bottom left corner. The paths are determined by specific movement rules; in this case, the only allowed moves are up or to the right. The grid is defined with specific coordinates: (0,0) is the bottom left and (5,10) is the top right corner. The goal of the exercise is to find all possible paths from the starting point to the destination using those movement rules.

Imagine you are trying to navigate a two-dimensional park layout. Starting from a picnic area at the bottom left corner of the park, you can only walk up towards the playground or right towards the fountain. The playground is right at the top right corner of the park, which you want to reach. Your challenge is to figure out the different routes you can take to reach the playground following the park's movement rules.

So, the question that we ask is how many different ways are going, are that go from (0, 0) to (5, 10). So, what do we mean by different ways. Well, here for instance is one grid path. This blue line takes us up from (0, 0), then right, then up, then right and so on. So, this traces out one particular sequence of edges along this grid taking us from the bottom left corner to the top right.

The paragraph discusses the specific task of counting the number of unique paths from the origin (0,0) to the target (5,10). A visual representation helps in understanding how these paths can differ. For example, one path could follow an up-right-up-right pattern, while others could vary the sequence of moves. The uniqueness of each path is based on the different combinations of 'up' and 'right' moves allowed, and this forms the basis for calculating the total number of possible paths.

Think of finding different routes to travel to a friend's house using different intersections. Each choice you make to either go straight (up) or turn (right) leads to a different route. Just as in the park example, where every decision at intersections creates a unique route to the playground, the paths on the grid represent the varied decisions you can take along the way.

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, right. There is no choice, I must walk 15 segments.

The solution to counting the grid paths is rooted in combinatorial theory. Moving from (0,0) to (5,10) requires a total of 15 movements: 5 to the right and 10 upwards. The principle here is that every path can be expressed as a combination of these moves, leading to the computation of how many ways we can organize these moves. This leads us to the classical combinatorial formula often used for counting paths in grids.

Imagine you have 15 steps to take to reach your friend's house – 10 of those steps must be straight up a street and 5 must be to the right to reach the driveway. The order in which you make these steps creates different routes to your friend's house. Thinking of those steps as a mixture of required actions gives you insight into how to calculate various routes you can take.

So, now if I take this and I think of this as an overall thing saying this is my 1st move, this is my 2nd move, this is my 3rd move and so on, up to the 15th move. Then if I tell you that you moved right, I moved right at these positions, at some five positions, then automatically I must have moved up at the remaining positions because I have to do exactly 5 and exactly 10 up.

This part outlines how to calculate the exact paths by determining the configurations of movements. By assigning each move a position in a sequence of 15 total moves, we can figure out where the right moves occur and determine the rest as up moves. The mathematical representation of this is known as 'combinations', specifically choosing positions for the right moves among the total moves.

Think of selecting specific days to take off work from a total of 15 weekdays where you need 5 days off. By setting which 5 specific days are your time-off days, the rest automatically become workdays, much like defining which movements on a grid make up your path while the others fulfill the requirement of reaching the destination.

Now, of course, instead of choosing the right positions I could also have told you the 10 positions where I moved up, that leaves 5 open positions where I. So, this would give us 15 choose 10, and so it is not a coincident that 15 choose 10 and 15 choose 5 are in fact the same expression.

The text introduces the alternative perspective of counting paths by selecting the positions for upward moves instead of rightward moves. This leads to the realization that both approaches yield the same mathematical result. This insight aligns with the fundamental properties of combinatorial mathematics, demonstrating that the total ways to arrange a set of items is invariant to which subset is selected.

If you think of a committee of 15 people where you can choose volunteers in any order, whether you pick 5 people to volunteer at a booth or decide which 10 will not, the total number of ways to form the committee remains constant regardless of the selection method.

So, the question now is, out of those 3003 paths that we said were there, from (0, 0) to (5, 10), how many of them are still valid if you are not allow to go through this intersection.

Here we shift our focus to scenarios where certain grid intersections are blocked. The task becomes identifying how many of the total paths exclude those blocked intersections. As some paths pass through these blocked areas, we need to account for and subtract those paths from the total valid paths originally counted.

Imagine one of the routes to the playground is closed due to construction. The original number of paths you could take is now affected; you need to calculate how many of those routes are still valid and can be used to reach your destination while avoiding closed paths.

So, it turns over 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, let me see how many ways are there of going through it and just remove them, right.

The concept discussed here is that if we can determine how many paths pass through a blocked intersection, we can simply subtract that from the total number of paths. This enables us to isolate valid paths that do not intersect with the blocked areas. There’s a combinatorial approach to calculating valid paths by breaking the journey into two parts separated by the blocked intersection.

Imagine a detour on your way to the playground due to construction. If you know how many routes went through the construction site, you can subtract those from your total routes to find out how many still remain valid and open for you to use.

Now, any part, which comes to the bottom 2 to 4 followed by any part, that goes from there to the top is a valid path passing through 2 to 4. So, I multiply these two numbers, I have to take 15 times 84 and again 1260.

As we look at multiple blocked areas, this principle of inclusion and exclusion becomes crucial. It helps calculate the paths that go through both blocked intersections, but we must account for any overlap in paths counted multiple times. Here, valid paths are calculated using combinations, identifying where paths intersect and adjusting counts to avoid duplicates.

Think of an event where students register for different subjects. If some are registered for multiple subjects, counting them simply by subject can lead to double counting. The inclusion-exclusion principle helps you calculate how many unique students are registered across multiple subject counts.

--