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Say there are two points $P_1(a_1,b_1)$ and $P_2(a_2,b_2)$, the number of ways of reaching $P_1$ from the origin is $w_1$ and $P_2$ from $P_1$ is $w_2$. (Here $a_1 and $b_1

$$w_1=\binom{a_1+b_1}{a_1},\quad w_2=\binom{(a_2-a_1)+(b_2-b_1)}{a_2-a_1}.$$

You get the above formula for $w_1$ by shifting to make $P_1$ the origin; the shift involves subtracting the coordinates of $P_1$ out of everything.

However: if the number of ways $W$ is given, how do we find a point $P_2$ such its distance from the origin is maximum?

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    @anon Is that clear?2012-01-29
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    The number of ways of reaching p2 through p1 is w1*w2; If there are w1 ways to reach point p1 and for each way, we have w2 ways to reach point p2. Thus w1*w22012-01-29
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    Don't you mean it's $w_1$ and not $p_1$ that is equal to $\displaystyle\frac{(a_1+b_1)!}{a_1! b_1!} = \ ^{a_1+b_1}C_{a_1} = \binom{a_1+b_1}{a_1}$, and similarly for $w_2$ instead of $p_2$? Apart from that, I think the question is clear as stated.2012-01-29
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    Not really, you still have a number of serious readability issues, but I think I can at least fix it now. (Also, sorry, I misread the definition $w_2$, you are correct about counting the ways.)2012-01-29
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    Alright, I've fixed it up and subsequently removed my downvote (which I did state was for unintelligibility; that comment is now removed). I did not vote to close. The combinatorics of two points seems irrelevant to your actual question though.2012-01-29
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    Thanks and thanks for editing the question. Actually I didn't know how to write those math equations. That's the reason why you guys had trouble.2012-01-29
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    The $P$ in your question is not mentioned anywhere before the last line, so it's not clear what it has to do with $W$. If what you are asking involves maximizing something over all $n,k$ with a given value of $n$-choose-$k$, then I think you'll find a very similar question about optimization involving binomial coefficients has been asked and answered here just in the last day.2012-01-29
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    @GerryMyerson can you give the link for that. Regarding the "P", call it P2, both are same2012-01-29
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    The question I had in mind was http://math.stackexchange.com/questions/103449/how-can-i-find-the-first-occurrence-of-a-number-in-pascals-triangle but see also the one in the answer of Jalaj.2012-01-29

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