document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=892694>Question 892694</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Is this a trick question or does the 6 seats in the first row alter the answer?\r<BR>\n" );
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document.write( "In how many ways can 5 students from a class of 16 be arranged in the first row of seats.  (There are 6 seats in the first row).\r<BR>\n" );
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document.write( "Help please. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #540720 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=540720\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "the number of ways you can get 5 students from 16 students is 16C5.\r<BR>\n" );
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document.write( "this does not take order into account.\r<BR>\n" );
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document.write( "this just gets you 5 students in each set without order.\r<BR>\n" );
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document.write( "then you want to arrange those 5 students in the 6 seats.\r<BR>\n" );
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document.write( "the number of possible sets of students is now 16C5 = 4368 possible sets of 5 that are each unique from each other.\r<BR>\n" );
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document.write( "each one of these sets needs to be arranged into the 6 seats.\r<BR>\n" );
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document.write( "take one at a time.\r<BR>\n" );
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document.write( "the vacant seat is equivalent to the 6th student that is added to each set.\r<BR>\n" );
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document.write( "therefore each set of 6 students can be arranged 6! times which is equal to 720 times.\r<BR>\n" );
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document.write( "the total number of possible arrangements will be 4368 * 720 = 3144960.\r<BR>\n" );
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document.write( "i tested this out with 2 out of 3 students being arranged in 3 seats.\r<BR>\n" );
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document.write( "it looks like the formula i gave you is correct.\r<BR>\n" );
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document.write( "here's the rationale.\r<BR>\n" );
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document.write( "the 3 students are abc (a is one of the students, b is one of the other students, c is the last of the 3 students).\r<BR>\n" );
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document.write( "the possible ways to get 2 out of the 3 students without order is 3C2 = 3.\r<BR>\n" );
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document.write( "those possible arrangements are:\r<BR>\n" );
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document.write( "ab<BR>\n" );
document.write( "ac<BR>\n" );
document.write( "bc\r<BR>\n" );
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document.write( "now each one of these sets of 2 students needs to be arranged in 3 seats.\r<BR>\n" );
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document.write( "that means one of the 3 seats will be vacant each time.\r<BR>\n" );
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document.write( "this effectively means that each set of 2 students will be arranged in 3! ways with the vacant seat representing the third student.\r<BR>\n" );
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document.write( "you have 3! ways for the first set of 2 students.<BR>\n" );
document.write( "you have 3! ways for the second set of 2 students.<BR>\n" );
document.write( "you have 3! ways for the third set of 2 students.\r<BR>\n" );
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document.write( "the total number of ways 2 students out of 3 can be arranged in 3 sets should be equal to 3C2 * 3! if this is correct.\r<BR>\n" );
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document.write( "that would be a total of 3*6 = 18 ways.\r<BR>\n" );
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document.write( "let's see if we can come up with those arrangements.\r<BR>\n" );
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document.write( "3 sets of students are ab, ac, bc.\r<BR>\n" );
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document.write( "we'll let v represent the vacant seat.\r<BR>\n" );
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document.write( "ab can be arranged in 3 seats in the following ways.\r<BR>\n" );
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document.write( "abv<BR>\n" );
document.write( "avb<BR>\n" );
document.write( "bav<BR>\n" );
document.write( "bva<BR>\n" );
document.write( "vab<BR>\n" );
document.write( "vba\r<BR>\n" );
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document.write( "that's 6 possible arrangements for the first set of 2 students.\r<BR>\n" );
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document.write( "the second set of students is ac<BR>\n" );
document.write( "they can be arranged in the following 6 ways.\r<BR>\n" );
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document.write( "acv<BR>\n" );
document.write( "avc<BR>\n" );
document.write( "cav<BR>\n" );
document.write( "cva<BR>\n" );
document.write( "vac<BR>\n" );
document.write( "vca\r<BR>\n" );
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document.write( "the third set of students is bc<BR>\n" );
document.write( "they can be arranged in the following 6 ways.\r<BR>\n" );
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document.write( "bcv<BR>\n" );
document.write( "bvc<BR>\n" );
document.write( "cbv<BR>\n" );
document.write( "cvb<BR>\n" );
document.write( "vbc<BR>\n" );
document.write( "vcb\r<BR>\n" );
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document.write( "looks like the formula is correct and should be able to be applied to your larger problem.\r<BR>\n" );
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document.write( "the solution to your larger problems is 16C5 * 6! = 3144960.\r<BR>\n" );
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document.write( "i won't swear this is correct, but it appears to be correct if my assumptions about what is being asked are correct.\r<BR>\n" );
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document.write( "here's another way to look at it.\r<BR>\n" );
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document.write( "if it was only 5 seats in the first row, the solution would have been 16P5 because then you would want the 5 students with order.\r<BR>\n" );
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document.write( "16P5 = 524160\r<BR>\n" );
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document.write( "this is the same answer you would get if you took 16C5 and then multiplied it by 5!.\r<BR>\n" );
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document.write( "you would get 16C5 * 5! = 4368 * 120 = 524160.\r<BR>\n" );
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document.write( "you're doing the same thing in this problem except you are multiplying by 6! rather than 5!.\r<BR>\n" );
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document.write( "the combination formula and the permutation formula are related in the following way.\r<BR>\n" );
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document.write( "nCx = nPx / x!\r<BR>\n" );
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document.write( "nPx = nCx * x!\r<BR>\n" );
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document.write( "this problem appears to be an adaptation of that second formula with the additional wrinkle that instead of multiplying by x!, you will be multiplying by (x+1)! that was required because of that extra seat that would be vacant each time.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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