document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1062843>Question 1062843</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An instructor gives an exam with fourteen questions. Students are allowed to choose any ten to answer.<BR>\n" );
document.write( "Suppose six questions require proof and eight do not.<BR>\n" );
document.write( "1-How many groups of ten questions contain at least one that requires proof?<BR>\n" );
document.write( "2-How many groups of ten questions contain at most three that require proof?\r<BR>\n" );
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document.write( "My answers<BR>\n" );
document.write( "for 1: C(6,2)*C(8,8)+C(6,3)*C(8,7)+C(6,4)*C(8,6)+C(6,5)*C(8,5)+C(6,6)*C(8,4)<BR>\n" );
document.write( "for 2: C(6,2)*C(8,8)+C(6,3)*C(8,7)<BR>\n" );
document.write( "i am not sure if they are correct </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #677919 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=677919\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> --<BR>\n" );
document.write( "For (1):  No matter how you select 10 questions from the 14, one or more of the 10 will have to require proof (only 8 do not require proof so there's always at least 2 that require proof).  Thus I think the answer is \"all possible selections\" and that is C(14,10).<BR>\n" );
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document.write( "For (2):  I'd reason as follows:  C(14,10) have at least 2 questions that require proof.   The maximum we care about is 3 questions requiring proof, so from the  C(14,10)  you'll need to subtract the number of groups with exactly {4 or 5 or 6} requiring proof.\r<BR>\n" );
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document.write( "C(6,4)*C(8,6) = # with 4 requiring proof<BR>\n" );
document.write( "C(6,5)*C(8,5) = # with 5 requiring proof<BR>\n" );
document.write( "C(6,6)*C(8,4) = # with 6 requiring proof\r<BR>\n" );
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document.write( "So I think (2) is:   C(14,10) - C(6,4)*C(8,6) - C(6,5)*C(8,5) - C(6,6)*C(8,4)<BR>\n" );
document.write( "-<BR>\n" );
document.write( "For #1, our answers agree C(14,10)=1001  and so does your answer.   Although we took different approaches, we both arrived at the same result.  It is likely you are correct.<BR>\n" );
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document.write( "For #2, our answers agree as well, value is 175 for both.<BR>\n" );
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document.write( "I can't explain why, but I see my way of looking at (2) as a \"take away\" problem while your solution is more of a \"constructive\" or additive solution.  Sometimes that happens I guess.   Anyway, good work!<BR>\n" );
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