document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=98520>Question 98520</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many 5 digit number exist between 10,500 and 11,000 if no digit may be repeated?\r<BR>\n" );
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document.write( "I answered 120 - 5x4x3x2x1 = 120 on a quiz and it was counted incorrect.  The teacher said that the answer was 210 but did not explain what I did wrong.  Help!  I'm trying to study for the test now and I can't figure it out. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #71683 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=71683\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's look at the first digit:\r<BR>\n" );
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document.write( "Since the number goes from 10,500 to 11,000, there is only one possibility for the first number: the only number it can be is \"1\"\r<BR>\n" );
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document.write( "Let's look at the second digit:\r<BR>\n" );
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document.write( "Since the number goes from 10,500 to 11,000, there are two  possibilities for the second number. The two possibilities are: 0 and 1\r<BR>\n" );
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document.write( "However since 1 is already taken, our only possibility for the second digit is 0\r<BR>\n" );
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document.write( "Let's look at the third digit:\r<BR>\n" );
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document.write( "Since the number goes from 10,500 to 11,000, we can have the numbers 5,6,7,8,9 (not zero since it's taken) for the third digit\r<BR>\n" );
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document.write( "So we have 5 possibilities for the third digit\r<BR>\n" );
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document.write( "Let's look at the fourth digit:\r<BR>\n" );
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document.write( "Now lets use any remaining digits to form the fourth possible digit. The remaining digits are:\r<BR>\n" );
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document.write( "2,3,4,5,6,7,8,9\r<BR>\n" );
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document.write( "Now lets say our third number is 5, so our possible numbers for the fourth digit is \r<BR>\n" );
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document.write( "2,3,4,6,7,8,9 (notice I've excluded 5)\r<BR>\n" );
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document.write( "So we'll have 7 possibilities\r<BR>\n" );
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document.write( "Let's look at the fifth digit:\r<BR>\n" );
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document.write( "Now lets use any remaining digits to form the fifth possible digit. The remaining digits are:\r<BR>\n" );
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document.write( "2,3,4,6,7,8,9\r<BR>\n" );
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document.write( "Now lets say our fourth number is 6, so our possible numbers for the fifth digit is \r<BR>\n" );
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document.write( "2,3,4,7,8,9 (notice I've excluded 6)\r<BR>\n" );
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document.write( "So we'll have 6 possibilities\r<BR>\n" );
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document.write( "Now multiply all of these combinations to get:\r<BR>\n" );
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document.write( "1*1*5*7*6=210\r<BR>\n" );
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document.write( "You can also look at it like this:\r<BR>\n" );
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document.write( "Since we're using a permutation, we can use nPr. So we have 7 possible numbers and we're choosing 3 digits. So we'll get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=nPr=%28n%21%29%2F%28n-r%29%21 ALIGN=MIDDLE  ALT=\"nPr=%28n%21%29%2F%28n-r%29%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given nPr formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7P3=%287%21%29%2F%287-3%29%21 ALIGN=MIDDLE  ALT=\"7P3=%287%21%29%2F%287-3%29%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in n=7 and r=3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7P3=%287%21%29%2F%284%21%29 ALIGN=MIDDLE  ALT=\"7P3=%287%21%29%2F%284%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7P3=%287%2A6%2A5%2A4%2A3%2A2%2A1%29%2F%284%2A3%2A2%2A1%29 ALIGN=MIDDLE  ALT=\"7P3=%287%2A6%2A5%2A4%2A3%2A2%2A1%29%2F%284%2A3%2A2%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Expand\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7P3=%287%2A6%2A5%2Across%284%2A3%2A2%2A1%29%29%2F%28cross%284%2A3%2A2%2A1%29%29 ALIGN=MIDDLE  ALT=\"7P3=%287%2A6%2A5%2Across%284%2A3%2A2%2A1%29%29%2F%28cross%284%2A3%2A2%2A1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Cancel like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7P3=7%2A6%2A5 ALIGN=MIDDLE  ALT=\"7P3=7%2A6%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7P3=210 ALIGN=MIDDLE  ALT=\"7P3=210\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Now multiply 7*6*5 to get 210\r<BR>\n" );
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document.write( "Notice how this is identical to the previous explanation (the numbers are just in different order) </SPAN>\n" );
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