document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=841513>Question 841513</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many different ways can 4 people be seated in a car with 7 seats? <BR>\n" );
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document.write( "This problem has been frustrating me for the past 10 minutes. It looks so simple yet I can't get it. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #507073 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=507073\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> my best guess is as follows:\r<BR>\n" );
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document.write( "driver's seat has 4 possibilities because there are 4 people and one of them has to be in the driver's seat.\r<BR>\n" );
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document.write( "that leaves 6 seats for 3 people.\r<BR>\n" );
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document.write( "looking at the first of the 6 seats, there are 3 choices there because there are 3 people left to choose that seat.<BR>\n" );
document.write( "looking at the second of the 6 seats, there are 2 choices there because there are 2 people left to choose that seat.<BR>\n" );
document.write( "looking at the third seat, there is 1 choice left there because there is one person looking for that seat.\r<BR>\n" );
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document.write( "so far you have 4 * 3! possibilities which is equal to 4 * 3 * 2 * 1 = 24.\r<BR>\n" );
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document.write( "now if you look at the 6 seats that aren't the driver's seat, only 3 out of the 6 have been chosen.\r<BR>\n" );
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document.write( "that, i believe, gets into how many sets of 3 can you make out of 6 where order is not important which is the combination formula.\r<BR>\n" );
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document.write( "that formula is 6C3 = (6*5*4) / (1*2*3) which results in 20 possible combinations.\r<BR>\n" );
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document.write( "your total number of possibilities is equal to 24 * 20 which would be equal to 480 possible arrangements.\r<BR>\n" );
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document.write( "this is extremely difficult to visualize.\r<BR>\n" );
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document.write( "i used much smaller examples to confirm the formula was reasonable.\r<BR>\n" );
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document.write( "i'll do one of them so you can see the logic i used.\r<BR>\n" );
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document.write( "assume 4 seats with 3 people.\r<BR>\n" );
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document.write( "number of possible choices for the driver's seat is 3.\r<BR>\n" );
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document.write( "that leaves 3 seats to be filled with the 2 remaining people.\r<BR>\n" );
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document.write( "the first seat has 2 choices and the second seat has 1 choice.\r<BR>\n" );
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document.write( "so far you have 3 * 2 * 1 = 6 possible choices.\r<BR>\n" );
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document.write( "since there are 3 seats and only 2 of them can be used at a time, the possible combinations of 2 seats at a time from 3 seats is 3C2 which is equal to 3.\r<BR>\n" );
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document.write( "your solution should be 6 * 3 = 18 possible ways the seats can be filled.\r<BR>\n" );
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document.write( "this is a big number but not that big that we can't confirm that the formula gave us the correct answer.\r<BR>\n" );
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document.write( "i'll do that below:\r<BR>\n" );
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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;\">\r\n" );
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document.write( "                         driver seat    seat 1     seat 2     seat 3\r\n" );
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document.write( "people are a, b, c\r\n" );
document.write( "x is the empty seat\r\n" );
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document.write( "1                             a            b          c          x\r\n" );
document.write( "2                             a            c          b          x\r\n" );
document.write( "3                             a            b          x          c\r\n" );
document.write( "4                             a            c          x          b\r\n" );
document.write( "5                             a            x          b          c\r\n" );
document.write( "6                             a            x          c          b\r\n" );
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document.write( "this is the first set with a in the driver's seat.\r\n" );
document.write( "the second set will have b in the driver's seat.\r\n" );
document.write( "the third set will have c in the driver's seat.\r\n" );
document.write( "the total should be 18.\r\n" );
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document.write( "i'll do the second set of 6 so you can see how it works.\r\n" );
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document.write( "7                             b            a          c          x\r\n" );
document.write( "8                             b            c          a          x\r\n" );
document.write( "9                             b            a          x          c\r\n" );
document.write( "10                            b            c          x          a\r\n" );
document.write( "11                            b            x          a          c\r\n" );
document.write( "12                            b            x          c          a\r\n" );
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document.write( "</PRE>\r<BR>\n" );
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document.write( "another 6 possible combinations will result from placing c in the driver's seat.\r<BR>\n" );
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document.write( "the total is 18.\r<BR>\n" );
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document.write( "the formula checks out.\r<BR>\n" );
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document.write( "i used it with 5 seats and believe it worked there as well unless i made a mistake.\r<BR>\n" );
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document.write( "that's why i think it's good.\r<BR>\n" );
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document.write( "so we'll do your problem again just to make sure it was calculated correctly.\r<BR>\n" );
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document.write( "you have 7 seats and 4 people.<BR>\n" );
document.write( "number of possible ways to fill the driver's seat is 4.<BR>\n" );
document.write( "number of possible ways to fill the first 3 seats is 3! = 6.<BR>\n" );
document.write( "only 3 of the 6 seats are filled.<BR>\n" );
document.write( "number of possible ways to fill the 6 seats 3 at a time is 6C3 = 20.<BR>\n" );
document.write( "total number of ways is 4 * 6 * 20 = 480.\r<BR>\n" );
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document.write( "without doing an exhaustive investigation i'm pretty sure the solutions is good.<BR>\n" );
document.write( "if you agree, then that's the way to go.\r<BR>\n" );
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