Question 13
A particle moves in the Cartesian plane from one lattice point to another according to the following rules:
• From any lattice point $(a, b),$ the particle may move only to $(a+1, b),$ $(a, b+1),$ or $(a+1, b+1).$
• 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 $(5, 5)?$
• From any lattice point $(a, b),$ the particle may move only to $(a+1, b),$ $(a, b+1),$ or $(a+1, b+1).$
• 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 $(5, 5)?$