document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179262>Question 1179262</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ten points in the plane are given, with no three collinear. Four<BR>\n" );
document.write( "distinct segments joining pairs of these points are chosen at random, all such segments<BR>\n" );
document.write( "being equally likely. The probability that some three of the segments form<BR>\n" );
document.write( "a triangle whose vertices are among the ten given points is m/n, where m and n<BR>\n" );
document.write( "are relatively prime positive integers. Find m + n. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850248 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850248\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step:\r<BR>\n" );
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document.write( "**1. Total Number of Segments:**\r<BR>\n" );
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document.write( "* With 10 points, the number of segments that can be formed is given by the combination formula:<BR>\n" );
document.write( "    * ¹⁰C₂ = 10! / (2! * 8!) = (10 * 9) / 2 = 45\r<BR>\n" );
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document.write( "**2. Total Number of Ways to Choose 4 Segments:**\r<BR>\n" );
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document.write( "* The number of ways to choose 4 segments from the 45 available is:<BR>\n" );
document.write( "    * ⁴⁵C₄ = 45! / (4! * 41!) = (45 * 44 * 43 * 42) / (4 * 3 * 2 * 1) = 148,995\r<BR>\n" );
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document.write( "**3. Number of Ways to Choose 4 Segments That DO NOT Form a Triangle:**\r<BR>\n" );
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document.write( "To find the probability of forming a triangle, it's easier to find the probability of *not* forming a triangle and subtract that from 1.\r<BR>\n" );
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document.write( "For 4 segments not to form a triangle, we need to consider the following cases:\r<BR>\n" );
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document.write( "* **Case 1: No intersections.** All four segments are completely separate.<BR>\n" );
document.write( "* **Case 2: One intersection.** Two segments intersect at a point, but no triangle is formed.<BR>\n" );
document.write( "* **Case 3: Two intersections.** Two sets of intersecting segments, but no triangle is formed.<BR>\n" );
document.write( "* **Case 4: All 4 segments form a quadrilateral.**\r<BR>\n" );
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document.write( "We will use the complementary counting method.\r<BR>\n" );
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document.write( "Consider the cases where we *cannot* form a triangle:\r<BR>\n" );
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document.write( "* **Case 1: All 4 segments are disjoint.** This is difficult to calculate directly.<BR>\n" );
document.write( "* **Case 2: Choose 4 segments such that no 3 form a triangle.**\r<BR>\n" );
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document.write( "Instead, let's look at the complementary case:\r<BR>\n" );
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document.write( "* **Case 1: 3 of the 4 segments form a triangle.**<BR>\n" );
document.write( "    * Choose 3 points out of 10 to form a triangle: ¹⁰C₃ = 120<BR>\n" );
document.write( "    * Choose 1 remaining segment: 42 segments can be chosen that don't form a triangle with the previous 3.<BR>\n" );
document.write( "    * So, 120 * 42 = 5040 ways to have 3 segments form a triangle, but we overcount, so we must divide by the number of times we can select the same triangle.<BR>\n" );
document.write( "* **Case 2: 4 segments form a complete quadrilateral.**<BR>\n" );
document.write( "    * Choose 4 points out of 10: ¹⁰C₄ = 210<BR>\n" );
document.write( "    * Each set of 4 points forms 3 possible quadrilaterals.<BR>\n" );
document.write( "    * 210 * 3 = 630 ways to have 4 segments form a quadrilateral.\r<BR>\n" );
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document.write( "Let's use a different approach.\r<BR>\n" );
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document.write( "We need to subtract the cases where we cannot form a triangle.\r<BR>\n" );
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document.write( "* **Case 1: 4 disjoint segments.**<BR>\n" );
document.write( "* **Case 2: 2 pairs of disjoint segments.**<BR>\n" );
document.write( "* **Case 3: A \"path\" of 4 segments.**<BR>\n" );
document.write( "* **Case 4: A \"star\" with 4 segments.**\r<BR>\n" );
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document.write( "Instead, let's find the number of ways to form a triangle.\r<BR>\n" );
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document.write( "* Choose 3 points out of 10: ¹⁰C₃ = 120<BR>\n" );
document.write( "* Choose 1 more segment from the remaining 42 segments: 42<BR>\n" );
document.write( "* Total ways to have a triangle: 120 * 42 = 5040\r<BR>\n" );
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document.write( "However, we are overcounting.<BR>\n" );
document.write( "We can select the 3 segments of the triangle from the chosen 4 segments in 4 ways.\r<BR>\n" );
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document.write( "So we have to divide by 4.\r<BR>\n" );
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document.write( "5040/4 = 1260\r<BR>\n" );
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document.write( "So there are at least 1260 ways.\r<BR>\n" );
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document.write( "Probability = 1260/148995 = 84/9933 = 28/3311.\r<BR>\n" );
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document.write( "m=28, n=3311\r<BR>\n" );
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document.write( "m+n = 3339\r<BR>\n" );
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document.write( "**Final Answer:**\r<BR>\n" );
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document.write( "m + n = 28 + 3311 = 3339<BR>\n" );
document.write( " </SPAN>\n" );
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