document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1087253>Question 1087253</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a student union meeting in a school, 16 students show up.Each shakes hands with each other exactly once.Determine the total number of handshakes.\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 #701522 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=701522\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">There are 16 students. Any given student has 15 people to shake hands with. Doing this 16 times means we have 16*15 = 240 orderings. \r<BR>\n" );
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document.write( "Keep in mind that instructions state <font color=blue>\"Each shakes hands with each other exactly once\"</font>. \r<BR>\n" );
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document.write( "So we need to get rid of the duplicate handshakes. For example, let's say we have student A and student B. One handshake is AB while another handshake is BA. The order doesn't matter as it's the same group. \r<BR>\n" );
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document.write( "To correct this, we divide by 2 and it gets us 240/2 = 120 which is the final answer.\r<BR>\n" );
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document.write( "Here is a table showing all the possible handshakes. Along the left and top borders are the student names (code name letters A through P, which is the 16th letter of the alphabet). \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"https://i.imgur.com/JVUD2ex.png\">\r<BR>\n" );
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document.write( "Combos like BB or CC are impossible so we ignore anything along the main diagonal. Those are marked with X. Anything below the diagonal is a mirror of what is above the diagonal. This is where the double-counting comes in. So we simply pick one side. I'm going to pick the stuff above the diagonal. Anything below the diagonal will have X in it. The boxes with S in it indicate a handshake. If you carefully count out the S letters, you'll find that there are 120 of them. An alternative is to sum the row counts to get: 15+14+13+...+3+2+1 and you'll get 15*(15+1)/2 = 15*16/2 = 120 which is the same as before. <BR>\n" );
document.write( "Using the formula is the preferred method since the table is a bit tedious. </font> </SPAN>\n" );
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