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avatar+3039 

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
avatar+7266 
+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
avatar+20011 
+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:

 

 

laugh

heureka  Feb 19, 2018
edited by heureka  Feb 19, 2018
 #3
avatar+3039 
+3

Thank you, both hectictar and heureka! Amazing explanations!

tertre  Feb 19, 2018
 #4
avatar+88898 
+3

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

 

 

cool cool cool

CPhill  Feb 19, 2018

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