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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.) So the number of ways (say $W$) of reaching $P_2$ from the origin through $P_1$ is $W=w_1\cdot w_2$. The number of combinations is given by

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

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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    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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Refer the answer given to Reverting the binomial coefficient. You may need to know Stirling's approximation, a related question for that can be found here and use of calculus or some clever trick later on.