document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=128922>Question 128922</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Manager of Muddlepool FC has a squad of 16 players including 2 goakeepelrs, 5 defenders, 5 midfeilders and 4 forwards. He plays a 4 4 2 formation. So must always have 1 goalkeeper, 4 defenders, 4 midfeilders, and 2 forward. All players can be chosen but they need to play a fair amount of games.<BR>\n" );
document.write( "How many different teams coud be chosen?<BR>\n" );
document.write( "In the squad are 4 brothers. 1 is a goalkeeper, 1 is a defender, 1 is a midfeilder and 1 is a forward. The brothers parents come only to a game with at least two of the brothers. The team thinks the parents will only watch half of the games but the brothers think it is more than half.<BR>\n" );
document.write( "In a season of 36 games how much more than half of the games are they likely to watch?<BR>\n" );
document.write( "How would it differ using a 4 3 3 formation? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #94439 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=94439\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Manager of Muddlepool FC has a squad of 16 players including 2 goalkeepers, 5 defenders, 5 midfeilders and 4 forwards. He plays a 4 4 2 formation. So must always have 1 goalkeeper, 4 defenders, 4 midfeilders, and 2 forward. All players can be chosen but they need to play a fair amount of games.<BR>\n" );
document.write( "How many different teams coud be chosen?<BR>\n" );
document.write( "# of ways to pick a goal keeper: 2<BR>\n" );
document.write( "# of ways to pick 4 defenders:5C4 = 5<BR>\n" );
document.write( "# of ways to pick 4 midfielders: 5C4 = 5<BR>\n" );
document.write( "# of ways to pick 2 forwards: 4C2 = 6<BR>\n" );
document.write( "Total # of teams = 2*5*5*6 = 300<BR>\n" );
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document.write( "\n" );
document.write( "In the squad are 4 brothers. 1 is a goalkeeper, 1 is a defender, 1 is a midfielder and 1 is a forward. The brothers parents come only to a game with at least two of the brothers. The team thinks the parents will only watch half of the games but the brothers think it is more than half.<BR>\n" );
document.write( "In a season of 36 games how much more than half of the games are they likely to watch?<BR>\n" );
document.write( "-----------<BR>\n" );
document.write( "Find # of teams possible with no brothers:<BR>\n" );
document.write( "Ways to pick goalkeeper: 1<BR>\n" );
document.write( "ways to pick defenders: 4C4 =1<BR>\n" );
document.write( "Ways to pick midfielders: 4C4=1<BR>\n" );
document.write( "ways to pick 2 forwards: 3C2 = 3<BR>\n" );
document.write( "# of teams possible with no brothers = 1*1*1*3 = 3<BR>\n" );
document.write( "---------------------<BR>\n" );
document.write( "Find # of teams possible with only one of the brothers:<BR>\n" );
document.write( "If goalkeeper is a brother # ways is : 1<BR>\n" );
document.write( "If defender cannot be brother # ways is 4C4=1<BR>\n" );
document.write( "If midfielder cannot be brother #ways is 4C4 = 1<BR>\n" );
document.write( "If forwards must not be one of the brothers 3C2 = 3<BR>\n" );
document.write( "----------<BR>\n" );
document.write( "Do the same making a defender one of the brothers and get 3 teams possible<BR>\n" );
document.write( "Do the same for midfielders and get 3 teams possible<BR>\n" );
document.write( "Do the same for forwards and get 1 team possible.<BR>\n" );
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document.write( "So there are a total of 3+3+3+3+1 = 13 teams possible with no brothers or<BR>\n" );
document.write( "one brother.<BR>\n" );
document.write( "Probability of a team having two or more of the brothers is 287/300<BR>\n" );
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document.write( "\n" );
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document.write( "In 36 games what is the probability 2 of the brothers are picked <BR>\n" );
document.write( "more than 18 times.  That is a binomial problem with n=36,p=287/300,18<x<=36<BR>\n" );
document.write( "Use 1-binomcdf(36,287/300,18) = practically one<BR>\n" );
document.write( "----------<BR>\n" );
document.write( "Conclusion: the probability is practically one that the parents will <BR>\n" );
document.write( "attend more than half the games<BR>\n" );
document.write( "----------------------------------<BR>\n" );
document.write( "How would it differ using a 4 3 3 formation?<BR>\n" );
document.write( "Available.........2.goalie.....5.defend.......5.middie......4.forward<BR>\n" );
document.write( "Needed............1............4..............3.............3......<BR>\n" );
document.write( "Brothers..........1............1..............1.............1......<BR>\n" );
document.write( "Not Bros..........1............4..............4.............3......<BR>\n" );
document.write( "# ways if no Bro..1...........4C4............4C3............3C3...= 3 teams<BR>\n" );
document.write( "# ways Goalie/Bro.ONE BRO.....4C4............4C3............3C3...= 4 teams<BR>\n" );
document.write( "# ways Defend/Bro.1...........ONE BRO........4C3............3C3...= 4 teams<BR>\n" );
document.write( "# ways Middie/Bro.1...........4C4............ONE BRO........3C3...= 1 team<BR>\n" );
document.write( "# ways Forwad/Bro 1...........4C4............4C3..........ONE BRO.= 4 teams<BR>\n" );
document.write( "-----------------------<BR>\n" );
document.write( "There are 16 teams that have no brother or one brother<BR>\n" );
document.write( "on them, from a total of 300 possible teams.<BR>\n" );
document.write( "Prob (team has at least two brothers on it) = 284/300<BR>\n" );
document.write( "Prob at least 18 of 36 games have two brothers = [1-binomcdf(36,284/300,18)]<BR>\n" );
document.write( "= approximately 1.<BR>\n" );
document.write( "==================================<BR>\n" );
document.write( "Cheers,<BR>\n" );
document.write( "Stan H. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );