document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1007417>Question 1007417</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An environmental organization has 19 members. Each member will be placed on exactly 1 of 4 teams. Each team will work on a different issue. The first team has 3 members, the second has 5, the third has 7, and the fourth has 4. In how many ways can these teams be formed? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #623348 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=623348\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i believe your solution is going to be:\r<BR>\n" );
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document.write( "19c3 * 16c5 * 11c7 * 4c4\r<BR>\n" );
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document.write( "ncx is the combination formula of n! / (x! * (n-x)!)\r<BR>\n" );
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document.write( "what happens, i believe, is this:\r<BR>\n" );
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document.write( "your first team is composed of 3 people taken from 19.\r<BR>\n" );
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document.write( "that gets you 19c3 = 969 possible teams.\r<BR>\n" );
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document.write( "for each of these teams, there are 16 members left that can be used to form the second team.\r<BR>\n" );
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document.write( "the second tesm is composed of 5 people taken from 16.\r<BR>\n" );
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document.write( "that gets you 16c5 = 4368 possible teams for EACH of the 969 first teams.\r<BR>\n" );
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document.write( "for each of these second teams, there are 11 members left that can be used to form the third team.\r<BR>\n" );
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document.write( "the third team is composed of 7 people taken from 11.\r<BR>\n" );
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document.write( "that gets you 11c7 = 330 possible teams for EACH of the 4368 second teams.\r<BR>\n" );
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document.write( "for each of the third teams, there are 4 members left that can be used to form the fourth team.\r<BR>\n" );
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document.write( "the fourth team is composed of 4 people taken from 4.\r<BR>\n" );
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document.write( "that gets you 4c4 = 1 possible team for EACH of the 330 third teams.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "19c3 * 16c5 * 11c7 * 4c4 becomes:\r<BR>\n" );
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document.write( "969 * 4368 * 330 * 1 = 1396755360 possible teams that can be formed.\r<BR>\n" );
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document.write( "does that make sense?\r<BR>\n" );
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document.write( "let's see with a little experiment.\r<BR>\n" );
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document.write( "suppose 4 teams are formed.\r<BR>\n" );
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document.write( "3 teams are composed of 2 members and the last team is composed of 1 member.\r<BR>\n" );
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document.write( "that would be 7 members forming 4 teams.\r<BR>\n" );
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document.write( "the team members are abcdefg.\r<BR>\n" );
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document.write( "the first team is formed as 7c2 = 21 possible teams.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "ab<BR>\n" );
document.write( "ac<BR>\n" );
document.write( "ad<BR>\n" );
document.write( "ae<BR>\n" );
document.write( "af<BR>\n" );
document.write( "ag<BR>\n" );
document.write( "bc<BR>\n" );
document.write( "bd<BR>\n" );
document.write( "be<BR>\n" );
document.write( "bf<BR>\n" );
document.write( "bg<BR>\n" );
document.write( "cd<BR>\n" );
document.write( "ce<BR>\n" );
document.write( "cf<BR>\n" );
document.write( "cg<BR>\n" );
document.write( "de<BR>\n" );
document.write( "df<BR>\n" );
document.write( "dg<BR>\n" );
document.write( "ef<BR>\n" );
document.write( "eg<BR>\n" );
document.write( "fg\r<BR>\n" );
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document.write( "now let's look at just one of these teams.\r<BR>\n" );
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document.write( "let's look at the team of ab.\r<BR>\n" );
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document.write( "when the team of ab is chosen, we have 5 members left that can form the second team of 2 members.\r<BR>\n" );
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document.write( "that second team can be formed from cdefg\r<BR>\n" );
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document.write( "the number of possible teams is 5c2 = 10.\r<BR>\n" );
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document.write( "the members of the second team are:\r<BR>\n" );
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document.write( "cd<BR>\n" );
document.write( "ce<BR>\n" );
document.write( "cf<BR>\n" );
document.write( "cg<BR>\n" );
document.write( "de<BR>\n" );
document.write( "df<BR>\n" );
document.write( "dg<BR>\n" );
document.write( "ef<BR>\n" );
document.write( "eg<BR>\n" );
document.write( "fg\r<BR>\n" );
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document.write( "now we'll look at the possible third teams that can be formed when just the first of the second teams has been formed.\r<BR>\n" );
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document.write( "ab are the chosen team from the first team.<BR>\n" );
document.write( "cd are the chosen team from the second team.\r<BR>\n" );
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document.write( "when ab and cd have been chosen for the first and second teams, the remaining members are efg\r<BR>\n" );
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document.write( "the third team can be formed in 3c2 possible ways.\r<BR>\n" );
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document.write( "those ways are:\r<BR>\n" );
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document.write( "ef<BR>\n" );
document.write( "eg<BR>\n" );
document.write( "fg\r<BR>\n" );
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document.write( "by now you should get the idea that the formula is good.\r<BR>\n" );
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document.write( "you get 7c2 * 5c2 * 3c2 * 1c1\r<BR>\n" );
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document.write( "what makes combinations and permutations so difficult is that it is very hard to prove the formulas are correct.\r<BR>\n" );
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document.write( "the way that i've found is to simplify the problem to the point where you can show that the formulas work.\r<BR>\n" );
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document.write( "i think i've done that here.\r<BR>\n" );
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document.write( "i'm reasonably confident the solution is correct.\r<BR>\n" );
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document.write( "if you know what the solution is supposed to be, you can check against it to see if you did it right.\r<BR>\n" );
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document.write( "o0therwise, there's no way of knowing unless you do something similar to what i've done above.\r<BR>\n" );
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document.write( "  </SPAN>\n" );
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