document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209003>Question 1209003</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose we want to choose 6 letters, without replacement, from 8 distinct letters.<BR>\n" );
document.write( "	<BR>\n" );
document.write( "	How many ways can this be done, if the order of the choices does not matter?\r<BR>\n" );
document.write( "\n" );
document.write( "How many ways can this be done, if the order of the choices matters? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847569 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=847569\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "This might be a bit strange, but I'll address the \"order matters\" portion first.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are n = 8 letters and r = 6 slots to fill.<BR>\n" );
document.write( "We have 8 choices for the first slot, then 7 for the next, and so on until reaching n-r+1 = 8-6+1 = 3 choices for the 6th slot.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiplying out these values gives: (8*7*6)*(5*4*3) = <font color=red>20160</font> different permutations where order matters.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Another way to reach this value is to use the nPr permutation formula.<BR>\n" );
document.write( "nPr = (n!)/( (n-r)! )<BR>\n" );
document.write( "which I'll let the student handle the scratch work if s/he chooses this route.<BR>\n" );
document.write( "The exclamation mark indicates a factorial.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now onto the \"order doesn't matter\" portion.<BR>\n" );
document.write( "For any group of 6 items, we have 6! = 6*5*4*3*2*1 = 720 ways to rearrange said group.<BR>\n" );
document.write( "This means that the previous value we found (20160) is too large by a factor of 720 if order doesn't matter.<BR>\n" );
document.write( "We'll divide the permutation value over r! = 6! = 720 to find the combination value.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A very useful formula is<BR>\n" );
document.write( "nCr = (nPr)/(r!)<BR>\n" );
document.write( "which is a way to connect permutations to combinations. \r<BR>\n" );
document.write( "\n" );
document.write( "There are 20160/720 = <font color=red>28</font> combinations.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This value can be found by looking at Pascal's Triangle. <BR>\n" );
document.write( "Since n = 8, we start with the row \"1,8,...\" <BR>\n" );
document.write( "r = 0 is the starting index and refers to the left-most value in this row. Increment r by 1 each time you move to the right. Once reaching r = 6, you should arrive at 28 in Pascal's Triangle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Or if you wanted you can use this nCr formula<BR>\n" );
document.write( "nCr = (n!)/( r!*(n-r)! )<BR>\n" );
document.write( "I'll let the student handle the scratch work if s/he chooses this route.<BR>\n" );
document.write( "Notice the presence of n! up top and (n-r)! down below; both of which are found in the nPr formula. <BR>\n" );
document.write( "So that can help explain why nCr = (nPr)/(r!) is true.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------------------------------<BR>\n" );
document.write( "--------------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answers:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "How many ways can this be done, if...<BR>\n" );
document.write( "Order doesn't matter?  <font color=red>28</font><BR>\n" );
document.write( "Order does matter?  <font color=red>20160</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );