1.2. Counting Paths without Blocks

Let us look at the problem of computing grid paths. We have 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 reach the top right corner.

This chunk introduces the concept of grid paths, where a traveler can move only up or right in a defined rectangular grid. The starting point is defined at the bottom left corner (0, 0), and the goal is to reach the top right corner, which corresponds to the coordinates of the grid. In this case, specifically, the grid has 5 columns and 10 rows, making the destination coordinates (5, 10). Understanding the constraints on movement is essential for calculating the different possible paths.

Imagine navigating a staircase where each step can only go up or move sideways. Just like you would figure out how many different footprints you could leave by either walking up or side-stepping at each point, this grid illustrates those choices in a two-dimensional space.

To go from (0,0) to (5,10), I must make a total of 15 steps: 5 right moves and 10 up moves.

In moving from the starting point to the destination, we must consider how many moves are needed in total. This movement totals to 15 steps, where 5 of those must be rightward moves and 10 must be upward moves. The total number of steps (15) is the sum of the segments needed to reach the desired position: 5 right moves + 10 up moves. Recognizing the quantity of each kind of move is a key step in determining the number of distinct paths available.

Think of this as planning a route in a city where you can either walk east (right) or north (up). If your destination is a specific point, you can count how many blocks you need to walk east versus how many blocks you need to walk north — this is similar to counting moves on a grid!

To find how many different paths there are, we need to choose 5 positions from the total 15 for the right moves. This is mathematically represented as 15 choose 5.

The calculation of paths can be handled using combinations, where we select positions for one type of move; in this case, the right moves among the total steps. This is represented as "15 choose 5", or mathematically as 15! / (5! * (15-5)!), which simplifies to 3003. This expression tells us the total number of unique ways to arrange these moves within the given constraints of the grid, demonstrating the fundamental principles of combinatorics.

Imagine you have a set of 15 toys and you need to select which 5 will go in your backpack. The different combinations of toys you can choose reflect similar choices as you plan your path on the grid.

If we have some intersections blocked, we need to calculate paths that avoid those blocks.

When intersections are blocked, not all paths computed previously will be valid anymore. We can systematically analyze this by calculating all paths that pass through the blocked intersection, then subtracting those from our original total. This approach allows us to still apply combinatorial reasoning but with a focus on eliminating invalid paths.

Think of a city where some roads are under construction and can't be taken. You would have to first figure out all the possible routes to your destination and then omit the routes that involve these closed roads to see which paths are still valid.

If multiple paths are blocked, we need to consider the intersection of the paths that pass through all blocked points.

When calculating paths through multiple blockages, we must consider the possibility of paths being counted more than once. This is managed by the inclusion-exclusion principle, which allows us to first subtract paths through each blocked intersection, and then add back those that were subtracted multiple times. This makes computation more precise and accounts for overlap in paths.

Consider counting students in various classes who are also involved in clubs. If some students are counted multiple times across different clubs, you would adjust your counts to avoid double-counting — similar to how we adjust our path counts with blockage intersections.

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