1

Previous question: link

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 $a_1<a_2$ and $b_1<b_2$.) 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?

Community
  • 1
Surya
  • 123
  • @anon Is that clear? – Surya Jan 29 '12 at 07:36
  • The number of ways of reaching p2 through p1 is w1w2; If there are w1 ways to reach point p1 and for each way, we have w2 ways to reach point p2. Thus w1w2 – Surya Jan 29 '12 at 07:37
  • 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. –  Jan 29 '12 at 07:44
  • 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.) – anon Jan 29 '12 at 07:49
  • 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. – anon Jan 29 '12 at 08:01
  • 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. – Surya Jan 29 '12 at 08:10
  • 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. – Gerry Myerson Jan 29 '12 at 08:43
  • @GerryMyerson can you give the link for that. Regarding the "P", call it P2, both are same – Surya Jan 29 '12 at 12:04
  • 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. – Gerry Myerson Jan 29 '12 at 23:55

1 Answers1

1

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.

Community
  • 1
Jalaj
  • 530