How manny ways

H

homeschool girl

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Marvin the fly starts at [MATH](0,0).[/MATH] Each step, Marvin moves one unit right or one unit up. He is trying to get to the point [MATH](5,7)[/MATH]. However, at [MATH](4,3)[/MATH] there is a frog that will eat him if he goes through that point. In how many ways can Marvin reach [MATH](5,7)[/MATH]?


I've tried [MATH]\frac{12!}{5!\cdot7!}-\frac{7!}{3!\cdot4!}[/MATH], but that was wrong.
 
I think you've got the right basic idea, but can you explain where the second term comes from? What is it supposed to count?

I think you're just missing one little piece, which I hope you'll discover as you explain your thinking.
 
the term I subtracted was the number of ways Marvin the fly can go to (4,3), I did the math like you would to count permutations of [MATH]rrrruuu[/MATH]
 
homeschool girl said:
Marvin the fly starts at [MATH](0,0).[/MATH] Each step, Marvin moves one unit right or one unit up. He is trying to get to the point [MATH](5,7)[/MATH]. However, at [MATH](4,3)[/MATH] there is a frog that will eat him if he goes through that point. In how many ways can Marvin reach [MATH](5,7)[/MATH]?
Click to expand...
Let's name points: \(S: (0,0),~F: (4,3)~\&~E: (5,7)\)
Now any paths \(S\to E\) consists of \(rrrrruuuuuuu\) in some order. That is where you got \(\dfrac{12!}{5!\cdot 7!}\) the total number of possible paths. But how many go thru \(F~?\) Those we want to remove. Well count the paths \(S\to F\) plus the paths \(F\to E\)
 
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You just counted the number of path to (4,3) but what about the number of paths from (4,3) to (5,7)?
 
Path from (0,0) to (4,3) is 7C4 and from (4,3) to (5,7) is 5C1

Now while I agree 7C4 + 5C1 = 35 + 5 = 40 why are you adding? I think that you took pka's hint to literally?
 
I am not sure what do you think? Whatever you decide can you please explain your reasoning. Thanks and please post back.
 
You have to decide if you add the 35 and 5 or if you multiply them. Which did you choose to do? Why did you choose that way?
 
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