Question

A particle moves in the Cartesian plane from one lattice point to another according to the following rules: 1. From any lattice point (a,b), the particle may move only to (a+1,b),(a,b+1), or (a+1,b+1). 2. There are no right angle turns in the particle's path. That is, the sequence of points visited contains neither a subsequence of the form (a,b),(a+1,b),(a+1,b+1) nor a subsequence of the form (a,b),(a,b+1),(a+1,b+1). How many different paths can the particle take from (0,0) to (5,5) ?

Answer 1

The number of different paths the particle can take from (0,0) to (5,5) is 252.

Step-by-step explanation:

The particle can only move to (a+1, b), (a, b+1), or (a+1, b+1) from any given lattice point (a, b). It cannot make right-angle turns in its path. To find the number of different paths it can take from (0,0) to (5,5), we can use combinatorics.

Starting from (0,0), we need to make a total of 5 moves in the x-direction and 5 moves in the y-direction to reach (5,5). We can arrange these moves in any order. Using combinatorics, the number of different paths is given by the binomial coefficient:

C(10, 5) = 252