document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=327022>Question 327022</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi I've been messing with this problem for so long now and I'm still not sure how to start with: There is a true-false quiz with 10 questions. How many possible answer keys are there if the answer to 6 questions are false and the other 9 questions are true?\r<BR>\n" );
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document.write( "In contrast, would I solve this question the same way: There is a multiple-choice test with 10 questions. 2 of the answers are A, 3 are B, 2 are C and 3 are D. How many possible answer keys are there? Thanks for the help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #234197 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20055)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=234197\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There is a true-false quiz with 10 questions. How many possible answer keys are there if the answer to 6 questions are false and the other 9 questions are true?<BR>\n" );
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document.write( "You mistyped since 10 is not 6 + 9.  The problem was either\r\n" );
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document.write( "1. There is a true-false quiz with 10 questions. How many possible answer keys\r\n" );
document.write( "are there if the answer to 6 questions are false and the other 4 questions are\r\n" );
document.write( "true?\r\n" );
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document.write( "or\r\n" );
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document.write( "2. There is a true-false quiz with 15 questions. How many possible answer keys\r\n" );
document.write( "are there if the answer to 6 questions are false and the other 9 questions are\r\n" );
document.write( "true?\r\n" );
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document.write( "I'll assume you meant 2:\r\n" );
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document.write( "Here is a typical answer key:\r\n" );
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document.write( "1   2   3   4   5   6   7   8   9  10  11  12  13  14  15\r\n" );
document.write( "T   F   F   T   T   T   F   T   F   T   T   F   T   F   T\r\n" );
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document.write( "We can work it either of two ways:\r\n" );
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document.write( "I. By placing all the T's first:\r\n" );
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document.write( "There are \"15 choose 6\" ways to choose the numbers to put T's.  There is\r\n" );
document.write( "only one way to fill in all the others, since they are all F's  So the answer\r\n" );
document.write( "is 15C6 = 5005 possible answer keys.\r\n" );
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document.write( " or\r\n" );
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document.write( "II. By placing all the F's first:\r\n" );
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document.write( "There are \"15 choose 9\" ways to choose the numbers to put F's.  There is\r\n" );
document.write( "only one way to fill in all the others, since they are all T's  So the answer\r\n" );
document.write( "is 15C9 = 5005 possible answer keys.\r\n" );
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document.write( "The answers are the same because 15C6 and 15C9 are the same. That's because the\r\n" );
document.write( "number of ways to choose 6 out of 15 is the same as the way to leave 9\r\n" );
document.write( "behind.\r\n" );
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document.write( "</PRE></b>\r<BR>\n" );
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document.write( "There is a multiple-choice test with 10 questions. 2 of the answers are A, 3<BR>\n" );
document.write( "are B, 2 are C and 3 are D. How many possible answer keys are there? <BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\"><b>\r\n" );
document.write( "We have 10 question numbers like this:\r\n" );
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document.write( "1   2   3   4   5   6   7   8   9  10  \r\n" );
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document.write( "There are \"10 choose 2\" or 10C2 question numbers to place the A's.\r\n" );
document.write( "Here is one of them:\r\n" );
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document.write( "1   2   3   4   5   6   7   8   9  10  \r\n" );
document.write( "        A               A         \r\n" );
document.write( "For each of the 10C2 ways to place the A's, in each case there will\r\n" );
document.write( "remain only 8 places to put the others. So, having placed the A's,\r\n" );
document.write( "there are \"8 choose 3\" or 8C3 question numbers to place the B's under.\r\n" );
document.write( "That's 10C2*8C3 ways to place the A's and B's.  Here is one of them:\r\n" );
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document.write( "1   2   3   4   5   6   7   8   9  10  \r\n" );
document.write( "B       A   B           A       B   \r\n" );
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document.write( "For each of the 10C2*8C2 ways to place the A's and B's, in each case \r\n" );
document.write( "there will remain only 5 places to put the others. So, having placed \r\n" );
document.write( "the A's and B's, there are \"5 choose 2\" or 5C2 question numbers to place\r\n" );
document.write( "the C's under.  That's 10C2*8C3*5C2 ways to place the A's, B's, and C's.\r\n" );
document.write( "Here is one of them:\r\n" );
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document.write( "1   2   3   4   5   6   7   8   9  10  \r\n" );
document.write( "B       A   B       C   A       B   C\r\n" );
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document.write( "For each of the 10C2*8C2*5C2 ways to place the A's, B's, and C's, in each\r\n" );
document.write( "case there will remain only 3 places to put the D's. So, having placed\r\n" );
document.write( "the A's, B's, and C's there are \"3 choose 3\" or 3C3 question numbers to place\r\n" );
document.write( "the D's under. [NOTICE THAT 3C3 IS JUST 1 because there is only one way to\r\n" );
document.write( "put D's under all 3].  That's 10C2*8C3*5C2*3C3 ways to place the A's, B's, C's,\r\n" );
document.write( "and D's.  That's 25200 ways.  Here is one of them completed\r\n" );
document.write( " \r\n" );
document.write( "1   2   3   4   5   6   7   8   9  10  \r\n" );
document.write( "B   D   A   B   D   C   A   D   B   C\r\n" );
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document.write( "Notice that it would not have mattered which letter we placed first.\r\n" );
document.write( "The numbers to multiply may be different, but the answer would always \r\n" );
document.write( "be the same.  Here are other possibilities of placing \r\n" );
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document.write( "10C2*8C3*5C2*3C3 =  45*56*10*1 = 25200, placing the A's, then B's, then C's, then D's\r\n" );
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document.write( "10C2*8C3*5C3*2C2 =  45*56*10*1 = 25200, placing the A's, then B's, then D's, then C's\r\n" );
document.write( " \r\n" );
document.write( "10C2*8C2*6C3*3C3 =  45*28*20*1 = 25200, placing the A's, then C's, then B's, then D's\r\n" );
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document.write( "10C3*7C2*5C3*2C2 = 120*21*10*1 = 25200, placing the B's, then A's, then D's, then C's\r\n" );
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document.write( "10C3*7C2*5C2*3C3 = 120*21*10*1 = 25200, placing the B's, then A's, then C's, then D's\r\n" );
document.write( " \r\n" );
document.write( "10C3*7C3*4C2*2C2 = 120*35*6*1  = 25200, placing the B's, then D's, then A's, then C's\r\n" );
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document.write( "Edwin</PRE>    </SPAN>\n" );
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