Can anybody explain to me the double counting method?

I was not able to understand it clearly in the brilliant blog, I want an explanation related to a real life situation. Can anybody explain it to me? Plzzzzzzzzzz

#BrilliantBlog #HelpMe! #Math

Note by Siddharth Kumar
8 years, 5 months ago

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4 votes

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Comments

Source: [IMO 1989/3]

Let nn and kk be positive integers, and let SS be a set of nn points such that :

a) No three points of SS are collinear,

b) For every point PP of SS, there are at least kk points of SS that are equidistant from PP.

Prove that k<12+2n k < \frac {1}{2} + \sqrt{2n} .

Calvin Lin Staff - 8 years, 5 months ago

Double Counting is a very subtle and yet useful method. I personally didn't understand how far its power reached until they made me face a difficult problem (I was told it was an IMO 3, but I have not confirmed it).

Let k,nk,n be natural numbers, and SS a set of nn points such that:

a) No three points are collinear.

b) For every point PP, there are at least kk points Q1,Q2,...,Qk Q_1, Q_2, ..., Q_k such that the distance PQ1=PQ2=...=PQkPQ_1 = PQ_2 = ... = PQ_k.

Prove that k<12+2n1 k < \frac 12 + \sqrt{2n-1} *

(*) I don't remember very well whether it was 2n22n-2 or 2n12n-1 or something like that.

This problem seems extremely difficult at first sight, and might seem rather Geometric than Combinatoric. However, the problem can be solved through double counting, but what can you double count?

Esteban Gomezllata - 8 years, 5 months ago

Well..frankly Calvin..how on earth do you know soo many stuffs?

Soham Chanda - 8 years, 5 months ago
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