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.

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

To start, let's visualize the given lattice path and set S:

(1,1) --(1,0)-- (2,1) --(1,-1)-- (3,0) --(0,1)-- (3,1) --(1,1)-- (4,2) --(0,1)-- (4,3) --(1,1)-- (5,4)

Set S: {(1,0), (0,1), (1,1), (1,-1), (-1,1)}

We can see that there are multiple possible paths from (1,1) to (2,4). To count the number of these paths, we can use dynamic programming.

Let's define a function P(x, y) that represents the number of paths in set S that end at the point (x, y). We need to find P(2, 4).

To determine P(x, y) for any point (x, y), we can use the following recursive relation:

P(x, y) = P(x-1, y) + P(x, y-1) - P(x-1, y-1)

Here's how we apply this relation:

1. Start by initializing P(1, 1) = 1, since it's the starting point.

2. Calculate P(2, 1) = P(1, 1) = 1, since we can only move right.

3. Calculate P(3, 1) = P(2, 1) = 1, since we can only move right.

4. Calculate P(3, 0) = P(2, 0) = 0, since moving down is not allowed.

5. Calculate P(4, 0) = P(3, 0) = 0, since moving down is not allowed.

6. Calculate P(4, 1) = P(3, 1) - P(3, 0) = 1 - 0 = 1, since moving down is not allowed.

7. Calculate P(4, 2) = P(3, 2) - P(3, 0) = 0 - 0 = 0, since moving down is not allowed.

8. Calculate P(5, 2) = P(4, 2) - P(4, 1) = 0 - 1 = -1, since there is no valid path.

9. Calculate P(5, 3) = P(4, 3) - P(4, 2) = 0 - 0 = 0, since moving down is not allowed.

10. Calculate P(5, 4) = P(4, 4) - P(4, 3) = 0 - 0 = 0, since moving down is not allowed.

Therefore, P(2, 4) = 0.

The total number of paths in set S that end at (2, 4) is 0.