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# counting problem

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Marvin the fly starts at (0,0).  Each step, Marvin moves one unit right or one unit up.  He is trying to get his home at (4,6).  However, at (2,4) here is a frog that will eat him if he goes through that point.  In how many ways can Marvin reach his home?

Dec 11, 2022

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Marvin has to take 4 + 6 = 10 steps to get to his home, and there are \({10 \choose 4} = 210\) ways to do this.

Now, we have to subtract the number of ways to get to (2, 4). There are \({6 \choose 2} = 15\) ways to do this.

So, there are \(210 - 15 = \color{brown}\boxed{195}\) ways to do this.

Dec 11, 2022