document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1098386>Question 1098386</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the design of a certain mechanical product, ten components are to be stacked into a cylindrical casing, in a manner that minimizes the impact of shocks. One end of the casing is designated as the top and the other end is the bottom.\r<BR>\n" );
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document.write( "a. If all components are different, how many different stacks are possible?<BR>\n" );
document.write( "b. If six components are identical to one another, but the others are different, how many different stacks are possible?<BR>\n" );
document.write( "c. If four components are of one type and identical to one another, and four components are of another type and identical to one another, but the others are different, how many different designs are possible? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #712781 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=712781\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. There are 10 choices for bottom piece.<BR>\n" );
document.write( "For each of those choices there are 9 for the piece to put on top of the first, and so on.<BR>\n" );
document.write( "The possibilities literally multiply.<BR>\n" );
document.write( "The total is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%21=10%2A9%2A8%2A%22...%22%2A3%2A2%2A1=highlight%283628800%29 ALIGN=MIDDLE  ALT=\"10%21=10%2A9%2A8%2A%22...%22%2A3%2A2%2A1=highlight%283628800%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "b. If 6 of the 10 pieces are identical, many of those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3628800 ALIGN=MIDDLE  ALT=\"3628800\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "assemblies will look exactly the same. They only differ in the order those 6 identical pieces were added.<BR>\n" );
document.write( "There are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%21=6%2A5%2A4%2A3%2A2%2A1=720 ALIGN=MIDDLE  ALT=\"6%21=6%2A5%2A4%2A3%2A2%2A1=720\"   DISABLED_style= \"max-width:90%; height:auto;\" > different orders,<BR>\n" );
document.write( "but if you cannot tell those 6 pieces apart,<BR>\n" );
document.write( "we count of just one design<BR>\n" );
document.write( "each set of 720 assemblies that differ only in the order the identical pieces were arranged.<BR>\n" );
document.write( "Then, there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%21%2F6%21=10%2A9%2A8%2A7=3628800%2F720=highlight%285040%29 ALIGN=MIDDLE  ALT=\"10%21%2F6%21=10%2A9%2A8%2A7=3628800%2F720=highlight%285040%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > different <BR>\n" );
document.write( "designs.<BR>\n" );
document.write( " <BR>\n" );
document.write( "c. If 4 pieces are of kind A, another 4 pieces are of kind B,<BR>\n" );
document.write( "and the remaining <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10-4-4=2 ALIGN=MIDDLE  ALT=\"10-4-4=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > pieces are different from all others,<BR>\n" );
document.write( "there when will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%21=4%2A3%2A2%2A1=24 ALIGN=MIDDLE  ALT=\"4%21=4%2A3%2A2%2A1=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways for the same design<BR>\n" );
document.write( "that differ only in the order the A pieces appear. And even when the orderof the A pieces is the same, there will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24 ALIGN=MIDDLE  ALT=\"24\"   DISABLED_style= \"max-width:90%; height:auto;\" > identical-looking design that differ only in the order the B pieces were arranged.<BR>\n" );
document.write( "All in all, there are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%21%2F%284%21%2A4%21%29=3628800%2F%2824%2A24%29=highlight%286300%29 ALIGN=MIDDLE  ALT=\"10%21%2F%284%21%2A4%21%29=3628800%2F%2824%2A24%29=highlight%286300%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > different-looking designs.<BR>\n" );
document.write( " </SPAN>\n" );
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