document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=209230>Question 209230</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many different three letter permutations can be formed from the letters in the word clipboard?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #158157 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=158157\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> clipboard<BR>\n" );
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document.write( "permutations are an ordered set.  this means that the same elements in a different order constitute a different permutation.<BR>\n" );
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document.write( "combinations are an unordered set.  this means that the same elements in a different order do not constitute a different combination<BR>\n" );
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document.write( "formulas:<BR>\n" );
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document.write( "number of permutations is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%29%21+%2F+%28n-x%29%21 ALIGN=MIDDLE  ALT=\"%28n%29%21+%2F+%28n-x%29%21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "number of combinations is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%29%21+%2F+%28%28x%29%21%2A%28n-x%29%21%29 ALIGN=MIDDLE  ALT=\"%28n%29%21+%2F+%28%28x%29%21%2A%28n-x%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "example 1<BR>\n" );
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document.write( "form sets of three letter combinations from the letters abcde.<BR>\n" );
document.write( "n = 5<BR>\n" );
document.write( "x = 3<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%29%21+%2F+%28%28x%29%21%2A%28n-x%29%21%29+=+%286%29%21+%2F+%28%283%29%21%2A%282%29%21%29+=+%285%2A4%2A3%2A2%2A1%29%2F%28%283%2A2%2A1%29%2A%282%2A1%29%29+=+%285%2A4%29%2F%282%2A1%29+=+10 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "they are:<BR>\n" );
document.write( "abc<BR>\n" );
document.write( "abd<BR>\n" );
document.write( "abe<BR>\n" );
document.write( "acd<BR>\n" );
document.write( "ace<BR>\n" );
document.write( "ade<BR>\n" );
document.write( "bcd<BR>\n" );
document.write( "bce<BR>\n" );
document.write( "bde<BR>\n" );
document.write( "cde<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "example 2<BR>\n" );
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document.write( "form sets of three letter permutations from the letters abcde.<BR>\n" );
document.write( "n = 5<BR>\n" );
document.write( "x = 3<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%29%21+%2F+%28%28n-x%29%21%29+=+%285%29%21+%2F+%28%282%29%21%29+=+%285%2A4%2A3%2A2%2A1%29%2F%28%282%2A1%29%29+=+%285%2A4%2A3%29+=+60 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "what is happening here is that for each of the combinations we calculated just above, we have 6 permutations.  10 * 6 = 60 permutations.<BR>\n" );
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document.write( "you get:<BR>\n" );
document.write( "abc * 6<BR>\n" );
document.write( "abd * 6<BR>\n" );
document.write( "abe * 6<BR>\n" );
document.write( "acd * 6<BR>\n" );
document.write( "ace * 6<BR>\n" );
document.write( "ade * 6<BR>\n" );
document.write( "bcd * 6<BR>\n" );
document.write( "bce * 6<BR>\n" );
document.write( "bde * 6<BR>\n" );
document.write( "cde * 6<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "if you look at each one of these combinations and make a permutation out of each one then you get the times 6 factor.<BR>\n" );
document.write( "take the combination abc for example:<BR>\n" );
document.write( "abc<BR>\n" );
document.write( "acb<BR>\n" );
document.write( "bac<BR>\n" );
document.write( "bca<BR>\n" );
document.write( "cab<BR>\n" );
document.write( "cba<BR>\n" );
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document.write( "the one combination of abc yielded 6 permutations of abc.<BR>\n" );
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document.write( "with the combination, all those additional permutations are not allowed because they are each composed of the same letters.  with the permutation, however, they are allowed because they are in a different order even though they contain the same combination of letters.<BR>\n" );
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document.write( "the formula for your problem is number of permutations which is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%29%21+%2F+%28n-x%29%21 ALIGN=MIDDLE  ALT=\"%28n%29%21+%2F+%28n-x%29%21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the word is clipboard<BR>\n" );
document.write( "n = 9<BR>\n" );
document.write( "x = 3<BR>\n" );
document.write( "n-x = 6<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "your formula becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289%29%21+%2F+%289-3%29%21 ALIGN=MIDDLE  ALT=\"%289%29%21+%2F+%289-3%29%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%21%2F6%21 ALIGN=MIDDLE  ALT=\"9%21%2F6%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > = (9*8*7*6*5*4*3*2*1)/(6*5*4*3*2*1) = 9*8*7 = 504<BR>\n" );
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document.write( "the number of combinations would be the number of permutations divided by x! which in this case is 3! which equals 6 making the number of combinations = 504 / 6 = 84.<BR>\n" );
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document.write( "one of those combinations is:<BR>\n" );
document.write( "cli<BR>\n" );
document.write( "this combination leads to 6 permutations which are:<BR>\n" );
document.write( "cli<BR>\n" );
document.write( "cil<BR>\n" );
document.write( "lic<BR>\n" );
document.write( "lci<BR>\n" );
document.write( "icl<BR>\n" );
document.write( "ilc<BR>\n" );
document.write( "------<BR>\n" );
document.write( "that happens for each of the 84 combinations to make the total number of permutations = 504.<BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );