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

https://latex.artofproblemsolving.com/2/e/a/2ea3e5319267c1bc86d52b268cd55cfdd5c411ba.png

tertre
Feb 19, 2018

#1**+3 **

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!

hectictar
Feb 19, 2018

#2**+3 **

**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:

heureka
Feb 19, 2018