Let S be the set of {(1,0),(0,1),(1,1),(1,−1),(−1,1)}-lattice path which begin at (1,1), do not use the same vertex twice, and never touch either the x-axis or the y-axis.

Let Px,y be the number of paths in S which end at the point (x,y). Determine P2,4.

Details and assumptions
A lattice path is a path in the Cartesian plane between points with integer coordinates.

A step in a lattice path is a single move from one point with integer coordinates to another.
The size of the step from (x1,y1) to (x2,y2) is (x2−x1,y2−y1).

The length of a lattice path is the number of steps in the path.

For a set S={(xi,yi)}ki=1, an S-lattice path is a lattice path where every step has size which is a member of S.

Well, let's see if I can come up with a humorous answer for you. P2,4 represents the number of paths in S that end at the point (2,4). To determine this, let me put on my clown glasses and do some mathemagical calculations.

Now, imagine a circus tent where the x-axis and the y-axis are tightropes, and our path is a daring tightrope walker. This walker starts at (1,1) and must perform a spectacular performance without ever touching the tightropes.

First, let's consider the possible steps the tightrope walker can take. We have five options: (1,0), (0,1), (1,1), (1,-1), and (-1,1). These steps are like the absurd tricks our clown performs to keep the audience entertained.

To reach (2,4), our clown must take four steps. But we have some conditions: no step can be repeated, and the walker can never touch the tightropes. So, let's calculate the number of possible paths that satisfy these conditions.

With my clownish calculations, I find that P2,4 is equal to the number of ways our tightrope walker can perform four unique, non-touching steps from (1,1) to (2,4), using our set of steps from S.

After juggling these numbers in my clown brain, I proudly present to you the answer:

P2,4 = 3

Hooray! Our tightrope walker has successfully reached (2,4) in three unique, non-touching steps. Bravo to the daring performer!

To determine P2,4, we need to count the number of paths in set S that end at the point (2,4). Let's break down the problem step-by-step.

Step 1: Determine the possible steps from (1,1)
We are given the set S = {(1,0),(0,1),(1,1),(1,-1),(-1,1)}. From (1,1), the possible steps are:
- Move right: (1,0)
- Move up: (0,1)
- Move diagonally: (1,1) or (1,-1)

Step 2: Count the possible paths to reach (2,4)
Starting from (1,1), we need to reach (2,4) without touching the x-axis or y-axis, and without revisiting any point.

As we can see, the only way to reach (2,4) is by taking the following steps:
- Move diagonally: (1,1)
- Move up: (0,1)
- Move up: (0,1)
- Move up: (0,1)
- Move right: (1,0)

Step 3: Count the number of possible paths
We have determined the sequence of steps to reach (2,4). Now, we need to count the number of distinct paths that can be formed from this sequence.

Since we have 5 steps, there are 5! = 5 factorial = 5 * 4 * 3 * 2 * 1 = 120 possible permutations of these steps.

However, we need to consider that we cannot revisit any point. Therefore, we need to adjust the count for the possible permutations that result in revisiting a point.

- We cannot revisit (1,1), so we have only 1 option for the first step.
- After the first step, we have 4 options.
- After the second step, we have 3 options.
- After the third step, we have 2 options.
- After the fourth step, we have only 1 option.

Therefore, the number of distinct paths from (1,1) to (2,4) is 1 * 4 * 3 * 2 * 1 = 24.

Hence, P2,4 = 24.

To determine P2,4, we need to count the number of paths in S that end at the point (2,4).

To do this, we can use a dynamic programming approach. We can build a grid, where each cell represents the number of paths from (1,1) to that cell. We will start by initializing the grid with all zeros.

Now, let's go through the lattice points in ascending order according to their x-coordinate and then their y-coordinate. For each lattice point (x,y), we will calculate the number of paths to that point based on the number of paths to its neighboring lattice points.

Since we cannot touch the x-axis or the y-axis, the only possible steps from (x,y) are either (x-1, y) or (x, y-1).

For the lattice point (1,1), we have 1 path.

For (2,1), there is only 1 possible step to reach (2,1) from (1,1), so the number of paths to (2,1) is the same as the number of paths to (1,1), which is 1.

For (1,2), there is also only 1 possible step to reach (1,2) from (1,1), so the number of paths to (1,2) is 1.

Now, for (2,2), we need to consider the paths that end at either (1,2) or (2,1). Since there are no loops allowed and we cannot visit the same vertex twice, the number of paths to (2,2) is the sum of the number of paths to (1,2) and the number of paths to (2,1), which is 1 + 1 = 2.

Following this pattern, we can continue to calculate the number of paths for the remaining lattice points until we reach (2,4).

Finally, we will have the value of P2,4, which represents the number of paths in S that end at (2,4).