document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201328>Question 1201328</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many different committees can be formed from 7 teachers and 43 students if the committee consists of 3 teachers and 3 ​students? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #835642 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=835642\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>431,935</font><BR>\n" );
document.write( "This number is slightly less than 432 thousand\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "There are 7 teachers to choose from for slot1.<BR>\n" );
document.write( "There are 6 teachers to choose from for slot2.<BR>\n" );
document.write( "There are 5 teachers to choose from for slot3.<BR>\n" );
document.write( "We count down by 1 each time we need to fill another slot. <BR>\n" );
document.write( "This is because we cannot reselect a certain teacher more than once.\r<BR>\n" );
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document.write( "There would be 7*6*5 = 42*5 = 210 permutations if the order of teachers mattered. <BR>\n" );
document.write( "However, the order doesn't matter on a committee. <BR>\n" );
document.write( "Each member has equal rank, and the seats aren't labeled. <BR>\n" );
document.write( "If we had seat names like \"chair person\", \"VP\", \"secretary\", \"treasurer\", etc, then order would matter.\r<BR>\n" );
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document.write( "Let's look at the number of ways to arrange a group of 3 people.<BR>\n" );
document.write( "I'll call them A,B,C<BR>\n" );
document.write( "We have 3*2*1 = 6 ways to do so as listed below.<ol><li>ABC</li><li>ACB</li><li>BAC</li><li>BCA</li><li>CAB</li><li>CBA</li></ol>Because there are 6 ways to rearrange a group of three people, we must divide the permutations result we got earlier by 6. This will give us how many combinations of teachers there are.\r<BR>\n" );
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document.write( "210/6 = 35<BR>\n" );
document.write( "This corrects the over-counting done by a factor of 6.\r<BR>\n" );
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document.write( "There are 35 ways to select the three teachers from a pool of seven. Order does not matter.\r<BR>\n" );
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document.write( "Here is another way to get the \"35\"<BR>\n" );
document.write( "n = 7 teachers<BR>\n" );
document.write( "r = 3 selections<BR>\n" );
document.write( "Use the nCr combination formula since order doesn't matter<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "7 C 3 = (7!)/(3!*(7-3)!)<BR>\n" );
document.write( "7 C 3 = (7!)/(3!*4!)<BR>\n" );
document.write( "7 C 3 = (7*6*5*4!)/(3!*4!)<BR>\n" );
document.write( "7 C 3 = (7*6*5)/(3!)<BR>\n" );
document.write( "7 C 3 = (7*6*5)/(3*2*1)<BR>\n" );
document.write( "7 C 3 = 210/6<BR>\n" );
document.write( "7 C 3 = 35<BR>\n" );
document.write( "Note: we have 7*6*5 = 210 up top and 3*2*1 = 6 down below<BR>\n" );
document.write( "These values were mentioned in the previous section.\r<BR>\n" );
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document.write( "Yet another way to get the 35 is to look at Pascal's Triangle<BR>\n" );
document.write( "Look at the row that has 1,7,... at the start<BR>\n" );
document.write( "The start index is r = 0 which corresponds to the \"1\" at the left edge.<BR>\n" );
document.write( "r = 1 then corresponds to the 7 and so on<BR>\n" );
document.write( "Count over four spaces to arrive at the index r = 3, and this value is the first copy of \"35\"\r<BR>\n" );
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document.write( "Unfortunately Pascal's triangle is not practical when determining how many ways there are to pick the three students from a pool of 43.<BR>\n" );
document.write( "It can be done, but the triangle would be really massive.\r<BR>\n" );
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document.write( "So I'll use the nCr formula<BR>\n" );
document.write( "n = 43 students<BR>\n" );
document.write( "r = 3 student selections<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "43 C 3 = (43!)/(3!*(43-3)!)<BR>\n" );
document.write( "43 C 3 = (43!)/(3!*40!)<BR>\n" );
document.write( "43 C 3 = (43*42*41*40!)/(3!*40!)<BR>\n" );
document.write( "43 C 3 = (43*42*41)/(3!)<BR>\n" );
document.write( "43 C 3 = (43*42*41)/(3*2*1)<BR>\n" );
document.write( "43 C 3 = 74046/6<BR>\n" );
document.write( "43 C 3 = 12341\r<BR>\n" );
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document.write( "Or a shortcut<BR>\n" );
document.write( "43*42*41 = 74046 permutations<BR>\n" );
document.write( "74046/6 = 12341 combinations<BR>\n" );
document.write( "We divide by 6 for the same reasoning as applied to the teachers.\r<BR>\n" );
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document.write( "Let's recap<ul><li>35 ways to pick the three teachers (order doesn't matter).</li><li>12341 ways to pick the three students (order doesn't matter).</li></ul>Therefore, we have 35*12341 = <font color=red>431,935</font> ways to pick the committee of 3 teachers and 3 students (order doesn't matter).<BR>\n" );
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