document.write( "Question 1143914: Four lines on a plane have at most N common points. FIND N.
\n" ); document.write( "A)7
\n" ); document.write( "B)6
\n" ); document.write( "C)5
\n" ); document.write( "D)4 \n" ); document.write( "
Algebra.Com's Answer #764871 by ikleyn(52803)\"\" \"About 
You can put this solution on YOUR website!
.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            More accurate formulation is THIS :\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "                Four \"highlight%28distinct%29\" lines on a plane have at most N common points. FIND N. \r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solution\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "    Two distinct straight lines may have at most 1 common point (the intersection point).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    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.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    More exactly, the maximum number of common points of N lines is equal to the number of all unordered pairs of lines of the given set of lines.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    In other words, the maximum number of common points of N lines is equal to the number of all combinations \r\n" );
document.write( "    of given lines taken 2 at a time  \"C%5Bn%5D%5E2\" = \"%28n%2A%28n-1%29%29%2F2\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    In case n = 4, the maximum number of common points of  lines is  \"%284%2A%284-1%29%29%2F2\" = \"%284%2A3%29%2F2\" = 2*3 = 6.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    This maximum number is achieved if and only if any two lines from the set intersect each other, i.e. are not parallel.\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solved, explained, answered and completed.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "/\/\/\/\/\/\/\/\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "I will give a BONUS problem to you to develop your mind.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "    Find the maximum possible number of intersection points that N circles may have on a plane ?\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "If you understand my solution to the previous problem, you should be able to solve this one, too.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-------------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "On Combinations,  see introductory lessons\r
\n" ); document.write( "\n" ); document.write( "    - Introduction to Combinations\r
\n" ); document.write( "\n" ); document.write( "    - PROOF of the formula on the number of Combinations\r
\n" ); document.write( "\n" ); document.write( "    - Problems on Combinations\r
\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of lessons on Permutations and Combinations\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic  \"Combinatorics: Combinations and permutations\". \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );