document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=452441>Question 452441</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are a number of people in a roo and each person shakes hands once with each of the other people.<BR>\n" );
document.write( "a) If there are 5 people in the room, how many handshakes have taken place?<BR>\n" );
document.write( "b) If there are 6 people in the room, how many handshakes have taken place?<BR>\n" );
document.write( "c) If there are 7 people in the room, how many handshakes have taken place?<BR>\n" );
document.write( "d) If there are n people in the room, how many handshakes have taken place? Write a recursive formula and explicit formula.<BR>\n" );
document.write( "e) Where are these numbers on Pascal's Triangle?<BR>\n" );
document.write( "f) If there were 190 handshakes altogether, how many people were in the room? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #310996 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=310996\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "So let's ay the room has only A and B---two people.\r<BR>\n" );
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document.write( "there is one handshake<BR>\n" );
document.write( "(A with B).\r<BR>\n" );
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document.write( "Three people (A, B, and C), there are three handshakes<BR>\n" );
document.write( "(A with B and C; B with C).\r<BR>\n" );
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document.write( "Now, let's try four people (A, B, C, and D), there are six handshakes<BR>\n" );
document.write( "(A with B, C, and D; B with C and D; C with D).\r<BR>\n" );
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document.write( "So if we have n-1 people, the number of handshakes is the sum of the first n numbers. This is n(n-1)/2\r<BR>\n" );
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document.write( "a) If there are 5 people in the room, how many handshakes have taken place?\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-1%29%2F2=5%285-1%29%2F2=5%2A4%2F2=5%2A2=10 ALIGN=MIDDLE  ALT=\"n%28n-1%29%2F2=5%285-1%29%2F2=5%2A4%2F2=5%2A2=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "b) If there are 6 people in the room, how many handshakes have taken place?\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-1%29%2F2=6%286-1%29%2F2=6%2A5%2F2=3%2A5=15 ALIGN=MIDDLE  ALT=\"n%28n-1%29%2F2=6%286-1%29%2F2=6%2A5%2F2=3%2A5=15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "c) If there are 7 people in the room, how many handshakes have taken place?\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-1%29%2F2=7%287-1%29%2F2=7%2A6%2F2=7%2A3=21 ALIGN=MIDDLE  ALT=\"n%28n-1%29%2F2=7%287-1%29%2F2=7%2A6%2F2=7%2A3=21\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "d) If there are n people in the room, how many handshakes have taken place? \r<BR>\n" );
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document.write( "Write a recursive formula and explicit formula.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-1%29%2F2 ALIGN=MIDDLE  ALT=\"n%28n-1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "e) Where are these numbers on Pascal's Triangle?\r<BR>\n" );
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document.write( "these are triangular numbers\r<BR>\n" );
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document.write( "f) If there were 190 handshakes altogether, how many people were in the room?\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-1%29%2F2=190%28190-1%29%2F2=190%2A189%2F2=17955 ALIGN=MIDDLE  ALT=\"n%28n-1%29%2F2=190%28190-1%29%2F2=190%2A189%2F2=17955\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Another way to think of this, if you have done some combinations, is for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > people, in how many ways can you pick two. This is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6C2+=+6%2A5%2F2=+15 ALIGN=MIDDLE  ALT=\"6C2+=+6%2A5%2F2=+15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=190 ALIGN=MIDDLE  ALT=\"190\"   DISABLED_style= \"max-width:90%; height:auto;\" > people<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=190C2+=+190%2A189%2F2=+17955 ALIGN=MIDDLE  ALT=\"190C2+=+190%2A189%2F2=+17955\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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