In a 6 by 6 grid, how many pathways can be drawn from S to F...?

[randy.lahey]

In a 6 by 6 grid, how many pathways can be drawn from S to F, (always moving to the right or down) that cross that main diagonal at least once? (You can use the Catalan sequence to answer this question too.) The diagram to the right illustrates one pathway that crosses the main diagonal twice. Don't try drawing all the pathways!

 

[pka]

In a 6 by 6 grid, how many pathways can be drawn from S to F, (always moving to the right or down) that cross that main diagonal at least once? (You can use the Catalan sequence to answer this question too.) The diagram to the right illustrates one pathway that crosses the main diagonal twice. Don't try drawing all the pathways!

Any path is made of six D's and six R's. The path shown in the diagram is \(\displaystyle DRRRDDDRDDRR\).

How many ways can we arrange six D's and six R's in a row?