document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194824>Question 1194824</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A man has a total of n friends; he wants to invite three of them for dinner every day during the first 4 months of the year 2023. Calculate the minimum number of friends , n, he must have if he does not want to invite the same combination of three friends more than once? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #827094 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=827094\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If we look at a calendar of 2023, then we'd find that:<ul><li>January has 31 days</li><li>February has 28 days (non leapyear)</li><li>March has 31 days</li><li>April has 30 days</li></ul>There are a total of 31+28+31+30 = 120 days in the first four months.\r<BR>\n" );
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document.write( "This must mean that there must be 120 or more combinations of three friends to invite friends over.\r<BR>\n" );
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document.write( "Recall that the nCr combinations can be found in Pascal's Triangle.\r<BR>\n" );
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document.write( "Go to this page here (or wherever your favorite resource is to look up Pascal's Triangle)<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.mathsisfun.com/pascals-triangle.html\">https://www.mathsisfun.com/pascals-triangle.html</a><BR>\n" );
document.write( "Scroll down until you reach the section titled \"The First 15 Lines\". <BR>\n" );
document.write( "You'll find a pretty massive triangle.\r<BR>\n" );
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document.write( "Looking through that triangle shows that 120 is buried inside it. <BR>\n" );
document.write( "It's found on the row that has 1, 10, 45, 120, ...\r<BR>\n" );
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document.write( "The second item of this row is the value of n we're after.\r<BR>\n" );
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document.write( "You should find that if n = 10 and r = 3, then nCr = 10C3 = 120\r<BR>\n" );
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document.write( "Scratch work:<BR>\n" );
document.write( "nCr = (n!)/(r!*(n-r)!)<BR>\n" );
document.write( "10C3 = (10!)/(3!*(10-3)!)<BR>\n" );
document.write( "10C3 = (10*9*8*7!)/(3!*7!)<BR>\n" );
document.write( "10C3 = (10*9*8)/(3!)<BR>\n" );
document.write( "10C3 = (10*9*8)/(3*2*1)<BR>\n" );
document.write( "10C3 = 720/6<BR>\n" );
document.write( "10C3 = 120\r<BR>\n" );
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document.write( "If the man has n = 10 friends, then he has 120 different trios of friends to invite over.<BR>\n" );
document.write( "This ensures that he has a different trio on every night for the first four months of 2023.\r<BR>\n" );
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document.write( "Keep in mind that repeated friends can be allowed (i.e. the same friend invited on multiple nights), just not repeated trios. <BR>\n" );
document.write( "For instance, let's say the man has ten friends with code names A,B,C,D,E,F,G,H,I,J<BR>\n" );
document.write( "One group to invite over is {A,B,C} for night one, then on night #2 could be the group {A,C,D}<BR>\n" );
document.write( "The group {A,C,B} is a repeat of the first trio, so it's not allowed. <BR>\n" );
document.write( "Order doesn't matter within any group.\r<BR>\n" );
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document.write( "If n > 10, then he'll have more possible trios.<BR>\n" );
document.write( "Therefore, n = 10 is the smallest possible value of n.\r<BR>\n" );
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document.write( "Answer: 10 friends.<BR>\n" );
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