Question 1143914
.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;More accurate formulation is <U>THIS</U> :


<pre>
                Four {{{highlight(distinct)}}} lines on a plane have at most N common points. FIND N. 
</pre>


<U>Solution</U>


<pre>
    Two <U>distinct</U> straight lines may have at most 1 common point (the intersection point).


    So the maximum number of common points of " n " lines is equal to the number of all pairs of lines of the given set of lines.


    More exactly, the maximum number of common points of N lines is equal to the number of all <U>unordered</U> pairs of lines of the given set of lines.


    In other words, the maximum number of common points of N lines is equal to the number of all combinations 
    of given lines taken 2 at a time  {{{C[n]^2}}} = {{{(n*(n-1))/2}}}.


    In case n = 4, the maximum number of common points of  lines is  {{{(4*(4-1))/2}}} = {{{(4*3)/2}}} = 2*3 = 6.


    This maximum number is achieved if and only if any two lines from the set intersect each other, i.e. are not parallel.
</pre>


Solved, explained, answered and completed.



/\/\/\/\/\/\/\/


I will give a <U>BONUS</U> problem to you to develop your mind.


<pre>
    Find the maximum possible number of intersection points that N circles may have on a plane ?
</pre>

If you understand my solution to the previous problem, you should be able to solve this one, too.


-------------------


On Combinations, &nbsp;see introductory lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Permutations/Introduction-to-Combinations-.lesson>Introduction to Combinations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Permutations/PROOF-of-the-formula-on-the-number-of-combinations.lesson>PROOF of the formula on the number of Combinations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Permutations/Problems-on-Combinations.lesson>Problems on Combinations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/OVERVIEW-the-lessons-on-Permutations-and-Combinations.lesson>OVERVIEW of lessons on Permutations and Combinations</A>

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic &nbsp;"<U>Combinatorics: Combinations and permutations</U>". 



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.