document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1152008>Question 1152008</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are five people in a room.  Each person shakes the hand of every other person exactly one. How many hands shakes are exchanged\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #773919 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=773919\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Here is a more sophisticated method for counting the number of handshakes; it is especially useful when the number of people is large.<br><BR>\n" );
document.write( "Suppose the number of people is n.  Then each of the n people shakes hands with each of the other (n-1) people, making a total of n*(n-1) handshakes.<br><BR>\n" );
document.write( "But when you count all of the handshakes by all the people in that way, you count each distinct handshake twice; so the actual number of handshakes is (n*(n-1))/2.<br><BR>\n" );
document.write( "In this problem, with 5 people, that gives you (5*(5-1))/2 = (5*4)/2 = 10.<br><BR>\n" );
document.write( "Of course, adding 4+3+2+1 is faster than that.... But if the number of people were 79 instead of 4, you wouldn't want to be adding 78+77+76+...+3+2+1 to find the number of handshakes.<br><BR>\n" );
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