document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=905735>Question 905735</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A quality control inspector samples 6 computer chips from a box containing 11 computer chips.  If the box contains 2 defective chips, in how many ways can he select a sample so that all 6 are good?<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 #549363 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=549363\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i believe your answer will be c(9,6) = (9*8*7*6*5*4) / 6*5*4*3*2*1) = (9*8*7) / (3*2*1) = (3*4*7) = (12*7) = 84\r<BR>\n" );
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document.write( "you have 11 to choose from but only 9 are good because 2 are bad.\r<BR>\n" );
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document.write( "the total possible ways you can get a sample is c(11,6) but the total possible ways you can get a sample where all are good is c(9,6).\r<BR>\n" );
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document.write( "i looked at this with much smaller numbers to see if i could come up with the formula.  \r<BR>\n" );
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document.write( "it was confirmed through several tries, the most complex being the one shown below:\r<BR>\n" );
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document.write( "assume a sample of 5 computer chips and 2 are defective.\r<BR>\n" );
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document.write( "you select a sample of 3.\r<BR>\n" );
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document.write( "in how many ways can you get all good?\r<BR>\n" );
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document.write( "the total number of possible ways to get any sample is c(5,3) = (5*4*3) / (3*2*1) = (5*4) / (2*1) = 10\r<BR>\n" );
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document.write( "the total number of possible ways to get all good is c(3,3) = 1.\r<BR>\n" );
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document.write( "if the formula is correct, there should only be one out of the 10 possible ways to get all three good.\r<BR>\n" );
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document.write( "assume the 5 chips are chip a,b,c,d,e\r<BR>\n" );
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document.write( "the total possible combinations of 3 are shown below.  there should be 10 of them.\r<BR>\n" );
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document.write( "assume d and e are defective.<BR>\n" );
document.write( "we'll mark the number of sets that contain either d or e in them with asterisks\r<BR>\n" );
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document.write( "abc<BR>\n" );
document.write( "abd ***<BR>\n" );
document.write( "abe ***<BR>\n" );
document.write( "acd ***<BR>\n" );
document.write( "ace ***<BR>\n" );
document.write( "ade ***<BR>\n" );
document.write( "bcd ***<BR>\n" );
document.write( "bce ***<BR>\n" );
document.write( "bde ***<BR>\n" );
document.write( "cde ***\r<BR>\n" );
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document.write( "the only set that didn't contain d or e in it was abc which is only 1 out of the 10 possible sets.\r<BR>\n" );
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document.write( "the formula was 3C3 instead of 5C3 because 2 were defective and we only wanted all good.\r<BR>\n" );
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document.write( "i applied the same logic to your problem, so instead of 11c6, you drew 9c6 which gave you all the possible combinations where all were good.\r<BR>\n" );
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document.write( "note that 9c6 and c(9,6) are two different ways to represent the same thing, namely how many different ways to draw sets of 6 from 9 where each set is different from the other and where order doesn't matter.\r<BR>\n" );
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