document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=66493>Question 66493</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this problem.\r<BR>\n" );
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document.write( "Toward the end of the principals' meeting for the Drug-Free Community campaign project, a total of 21 handshakes were exchanged. Assuming each principal shakes hands once with the other principals, how many principals were present? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #47173 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=47173\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Picture the group of principals in a room. <BR>\n" );
document.write( "Pick one principal to shake all the others hands.<BR>\n" );
document.write( "He will shake n-1 hands, where  n = himself plus<BR>\n" );
document.write( "all the others.<BR>\n" );
document.write( "The next principal doesn't need to shake hands<BR>\n" );
document.write( "again with the 1st one, so he shakes n-2 hands.<BR>\n" );
document.write( "The next principal shakes n-3 hands, and so on.<BR>\n" );
document.write( "The total number of handshakes is 21, so<BR>\n" );
document.write( "(n-1) + (n-2) + (n-3) + . . . = 21<BR>\n" );
document.write( "Try different values of n that will not lead to a result<BR>\n" );
document.write( "less than 21 or a negative result<BR>\n" );
document.write( "How about n = 1? That's 0 handshakes.<BR>\n" );
document.write( "How about n=2<BR>\n" );
document.write( "(2-1) + (2-2) = 1 Just 1 handshake<BR>\n" );
document.write( "n = 3<BR>\n" );
document.write( "(3-1) + (3-2) + (3-3) = 3 handshakes<BR>\n" );
document.write( "n = 4<BR>\n" );
document.write( "(4-1) + (4-2) + (4-3) + (4-4) = 6 handshakes<BR>\n" );
document.write( "n = 5<BR>\n" );
document.write( "(5-1) + (5-2) + (5-3) + (5-4) + (5-5) = 10 handshakes<BR>\n" );
document.write( "n = 6<BR>\n" );
document.write( "(6-1) + (6-2) + (6-3) + (6-4) + (6-5) + (6-6) = 15 handshakes<BR>\n" );
document.write( "n = 7<BR>\n" );
document.write( "(7-1) + (7-2) + (7-3) + (7-4) + (7-5) + (7-6) + (7-7) = 21 handshakes<BR>\n" );
document.write( "So, there are 7 principals   </SPAN>\n" );
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