help!!!!!1

Here is my guess...

From  E  to  F  there are  4  different paths:

South, East, East, East

East, South, East, East

East, East, South, East

East, East, East, South

From  F  to  G  there are  10  different paths:

South, South, South, East, East

South, South, East, South, East

South, South, East, East, South

South, East, South, South, East

South, East, South, East, South

South, East, East, South, South

East, South, South, South, East

East, South, South, East, South

East, South, East, South, South

East, East, South, South, South

4 * 10  =  40

I think there are  40  different 9-step paths from  E  to  G  which pass through  F .

If this is right, maybe someone else can give a better explanation!

How many 9 step paths are there from E to G which pass through F?

We can simplify:

\(\begin{array}{|rcll|} \hline && \dfrac{4!}{3!1!}\times \dfrac{5!}{2!3!} \\\\ &=& \dfrac{3!4}{3!1!}\times \dfrac{3!\times 4 \times 5}{2!3!} \\\\ &=& \dfrac{ 4}{ 1!}\times \dfrac{4 \times 5}{2! } \\\\ &=& \dfrac{ 4}{1}\times \dfrac{4 \times 5}{2} \\\\ &=& 4\times 2 \times 5 \\\\ &=& 4\times 10 \\ &=& 40 \\ \hline \end{array}\)

There are 40
9 step paths are there from E to G which pass through F.

In general you can solve 2-D Pathways:

Thank you, both hectictar and heureka! Amazing explanations!

Thanks, hectictar and heureka.....very informative!!!!