1.1. Introduction to 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). And the question that we ask is how many different ways are going, are that go from (0, 0) to (5, 10).

This chunk introduces the concept of grid paths by describing a rectangular grid where movement is restricted to going up or right. Starting at the bottom-left corner (0,0) and aiming to reach the top-right corner (5,10) poses a question about how many distinct paths one can take to reach that destination given the movement constraints.

Think of navigating a city where you can only walk north or east. You start at your house (coordinates 0,0) and want to reach a friend's house located northeast (5 blocks north and 10 blocks east). The challenge is to find different routes you could take without backtracking.

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.

This chunk explains that the problem of counting paths can be solved using combinatorial analysis. To get from (0, 0) to (5, 10), you need to make exactly 15 moves: 5 moves right and 10 moves up. These movements can be arranged in various sequences, with the total number of moves being fixed at 15, leading to various combinations.

Imagine you have a puzzle with tiles: you can either move a tile right or pull it up. For every completed row of 5 tiles, you can only move up to the next row after finishing. If you have a fixed number of moves (5 right and 10 up), think of it as arranging a recipe where you choose the order in which you add ingredients.

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 5 right and 10 up.

This part elaborates on how paths are formed by selecting specific moves among the total steps. It emphasizes that knowing the positions of the right moves inherently dictates the positions of the up moves. Mathematically, this is expressed using the binomial coefficient, often noted as 'n choose k' (or C(n, k)), which calculates how many ways we can choose k elements from a set of n.

Think of organizing a concert where you know you have 15 time slots but need to decide 5 for solo acts and 10 for group performances. By picking the slots for the solo acts, the remaining slots automatically belong to group performances, similar to arranging a mixture of performances in a set time.

So, the blue path that we had drawn before actually goes through this intersection. So, this path is no longer a valid path. The red path is ok, because it bypasses it, but the yellow path unfortunately also goes through this intersection. So, of the three paths, that we had seen so far, two actually do not go through. 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 allowed to go through this intersection.

Here, the concept shifts to identifying valid paths when certain intersections are blocked. The paths that intersect with the blocked location are no longer valid. So, from the original count of 3003 paths, one must determine how many do not include paths that pass through the blocked intersection.

Imagine a maze where some routes are blocked by walls. You previously identified all possible paths through the maze, but now you need to recalculate which paths remain open when certain exits are sealed off.

Now, if I count all the paths going through (2,4), which I have already done, I can subtract this. Similarly, I can compute all the paths going through (4,4) and subtract those. 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.

This chunk introduces the inclusion-exclusion principle, which is used when accounting for multiple blocked paths. If two paths block different points, removing those paths may inadvertently eliminate certain valid paths multiple times. Hence, it's crucial to add back paths counted twice to rectify the total count of valid paths.

Think of counting students who play multiple sports: if you count those who play soccer and rugby separately, some students who play both might end up counted twice. To get an accurate count of distinct athletes, you need to account for those double-counted.

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