document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204436>Question 1204436</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A six-team hockey league has a schedule requiring each team to play each of the other teams 9 times. How many games will be played in total?  </SPAN>\n" );
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document.write( "Let's consider the scenario that each team plays another team exactly once.\r<BR>\n" );
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document.write( "Label the 6 teams as A,B,C,D,E,F\r<BR>\n" );
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document.write( "Notation like AB means teams A and B play together.\r<BR>\n" );
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document.write( "Here are the outcomes for team A<BR>\n" );
document.write( "AB, AC, AD, AE, AF\r<BR>\n" );
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document.write( "Here's team B's outcomes<BR>\n" );
document.write( "BC, BD, BE, BF<BR>\n" );
document.write( "We don't list BA because AB was already covered.\r<BR>\n" );
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document.write( "Keep this process going and we have this full schedule:<BR>\n" );
document.write( "AB, AC, AD, AE, AF<BR>\n" );
document.write( "BC, BD, BE, BF<BR>\n" );
document.write( "CD, CE, CF<BR>\n" );
document.write( "DE, DF<BR>\n" );
document.write( "EF<BR>\n" );
document.write( "There's a staircase pattern going on.<BR>\n" );
document.write( "There are 5 items in row 1, 4 in row 2, etc.<BR>\n" );
document.write( "5+4+3+2+1 = 15 games in the season if each team plays once against each other.\r<BR>\n" );
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document.write( "What's another way to arrive at 15?<BR>\n" );
document.write( "Well consider two slots for the two teams chosen to play together. <BR>\n" );
document.write( "We have 6 choices for the first slot and 5 for the next.<BR>\n" );
document.write( "That's 6*5 = 30 permutations and 30/2 = 15 combinations. <BR>\n" );
document.write( "We divide by 2 so we avoid double-counting. Something like AB is the same as BA.<BR>\n" );
document.write( "Another way to arrive at the 15 is to use the nCr combination formula. Plug in n = 6 and r = 2.<BR>\n" );
document.write( "And yet another way to get 15 is to look at Pascal's Triangle. Look at the row that starts with 1,6,... and count 2 spots over to land on 15.\r<BR>\n" );
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document.write( "As you can see, there are plenty of options here.\r<BR>\n" );
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document.write( "Since there are 15 different matchups when each team plays another team once, there must be 9*15 = 135 different games if each team plays another team 9 times.\r<BR>\n" );
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document.write( "Answer: <font color=red>135 games</font><BR>\n" );
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