document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1152786>Question 1152786</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A committee of 6 U.S. senators is to be formed with 3 Democrats and 3 Republicans. In how many ways can this be done if there are 61 Democratic senators and 39 Republican senators?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #774858 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=774858\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "We will use the combination function<BR>\n" );
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document.write( "to count the number of ways to select members. We use the combination function (not the permutation function) because order does not matter when selecting a committee. Any member does not outrank another, and there are no special positions. All that matters is the group as a whole rather an any particular individual. \r<BR>\n" );
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document.write( "Your book may use notation that looks like <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large _n C_r\"> instead of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Cr%29 ALIGN=MIDDLE  ALT=\"C%28n%2Cr%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but its the same thing. \r<BR>\n" );
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document.write( "The exclamation marks represent factorials. Writing something like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%21 ALIGN=MIDDLE  ALT=\"6%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%21+=+6%2A5%2A4%2A3%2A2%2A1 ALIGN=MIDDLE  ALT=\"6%21+=+6%2A5%2A4%2A3%2A2%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > so we start at 6 and count our way down to 1 multiplying along the way.\r<BR>\n" );
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document.write( "Note how something like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%21+=+4%2A3%2A2%2A1 ALIGN=MIDDLE  ALT=\"4%21+=+4%2A3%2A2%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is fully contained inside of the expression for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%21 ALIGN=MIDDLE  ALT=\"6%21\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This means, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%21+=+6%2A5%2A4%2A3%2A2%2A1+=+6%2A5%2Ahighlight%284%2A3%2A2%2A1%29+=+6%2A5%2Ahighlight%284%21%29+=+6%2A5%2A4%21 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In short, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%21+=+6%2A5%2A4%21 ALIGN=MIDDLE  ALT=\"6%21+=+6%2A5%2A4%21\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This will come in handy later on. Specifically the lines where I write \"<font color=blue>use the factorial trick</font>\" (see below). This is to cancel out factorials that would otherwise be massively large numbers.\r<BR>\n" );
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document.write( "For the Democrats, we have n = 61 people to choose from and r = 3 slots to fill. <BR>\n" );
document.write( "Use the combination function to figure out how many ways there are to do this.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Cr%29+=+%28n%21%29%2F%28r%21%2A%28n-r%29%21%29 ALIGN=MIDDLE  ALT=\"C%28n%2Cr%29+=+%28n%21%29%2F%28r%21%2A%28n-r%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2861%2C3%29+=+%2861%21%29%2F%283%21%2A%2861-3%29%21%29 ALIGN=MIDDLE  ALT=\"C%2861%2C3%29+=+%2861%21%29%2F%283%21%2A%2861-3%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace n with 61, replace r with 3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2861%2C3%29+=+%2861%2A60%2A59%2A58%21%29%2F%283%21%2A58%21%29 ALIGN=MIDDLE  ALT=\"C%2861%2C3%29+=+%2861%2A60%2A59%2A58%21%29%2F%283%21%2A58%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <font color=blue>Use the factorial trick</font>. We started at 61 and stop at 58 because of the 58! in the denominator.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2861%2C3%29+=+%2861%2A60%2A59%2Ahighlight%2858%21%29%29%2F%283%21%2Ahighlight%2858%21%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > We have these highlighted terms pair up\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2861%2C3%29+=+%2861%2A60%2A59%2Across%2858%21%29%29%2F%283%21%2Across%2858%21%29%29 ALIGN=MIDDLE  ALT=\"C%2861%2C3%29+=+%2861%2A60%2A59%2Across%2858%21%29%29%2F%283%21%2Across%2858%21%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and cancel out. A much simpler expression is left over \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2861%2C3%29+=+%2861%2A60%2A59%29%2F%283%2A2%2A1%29 ALIGN=MIDDLE  ALT=\"C%2861%2C3%29+=+%2861%2A60%2A59%29%2F%283%2A2%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Expand out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%21 ALIGN=MIDDLE  ALT=\"3%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A2%2A1 ALIGN=MIDDLE  ALT=\"3%2A2%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "At this point, we have three numbers multiplied in the numerator. It is not a coincidence this lines up with r = 3. So another shortcut is to start with the number n = 61 and count down by 1 until you have r = 3 items to multiply out in the numerator. Then you divide over 3! = 3*2*1\r<BR>\n" );
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document.write( "Let's simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2861%2C3%29+=+35990 ALIGN=MIDDLE  ALT=\"C%2861%2C3%29+=+35990\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "There are 35990 ways to pick the Democrats.\r<BR>\n" );
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document.write( "We'll use this number later, so let A = 35990\r<BR>\n" );
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document.write( "Repeat for the Republicans\r<BR>\n" );
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document.write( "n = 39\r<BR>\n" );
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document.write( "r = 3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2C+r%29+=+%28n%21%29%2F%28+r%21%2A%28n-r%29%21+%29 ALIGN=MIDDLE  ALT=\"C%28n%2C+r%29+=+%28n%21%29%2F%28+r%21%2A%28n-r%29%21+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2839%2C+3%29+=+%2839%21%29%2F%28+3%21%2A36%21+%29 ALIGN=MIDDLE  ALT=\"C%2839%2C+3%29+=+%2839%21%29%2F%28+3%21%2A36%21+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2839%2C+3%29+=+%2839%2A38%2A37%2A36%21%29%2F%28+3%21%2A36%21+%29 ALIGN=MIDDLE  ALT=\"C%2839%2C+3%29+=+%2839%2A38%2A37%2A36%21%29%2F%28+3%21%2A36%21+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <font color=blue>Use the factorial trick</font>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2839%2C+3%29+=+%2839%2A38%2A37%2Ahighlight%2836%21%29%29%2F%28+3%21%2Ahighlight%2836%21%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Terms pair up\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2839%2C+3%29+=+%2839%2A38%2A37%2Across%2836%21%29%29%2F%28+3%21%2Across%2836%21%29+%29 ALIGN=MIDDLE  ALT=\"C%2839%2C+3%29+=+%2839%2A38%2A37%2Across%2836%21%29%29%2F%28+3%21%2Across%2836%21%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and cancel out\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2839%2C+3%29+=+%2839%2A38%2A37%29%2F%28+3%21+%29 ALIGN=MIDDLE  ALT=\"C%2839%2C+3%29+=+%2839%2A38%2A37%29%2F%28+3%21+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since r = 3, we have 3 numbers multiplied in the numerator. We started at n = 39 and counted down by 1 each time.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2839%2C+3%29+=+%2839%2A38%2A37%29%2F%28+3%2A2%2A1+%29 ALIGN=MIDDLE  ALT=\"C%2839%2C+3%29+=+%2839%2A38%2A37%29%2F%28+3%2A2%2A1+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Expand out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%21 ALIGN=MIDDLE  ALT=\"3%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A2%2A1 ALIGN=MIDDLE  ALT=\"3%2A2%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2839%2C+3%29+=+9139 ALIGN=MIDDLE  ALT=\"C%2839%2C+3%29+=+9139\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let B = 9139 represent the number of ways to pick the Republicans.\r<BR>\n" );
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document.write( "To summarize what we found so far,<BR>\n" );
document.write( "A = 35990 is the number of ways to pick the Democrats<BR>\n" );
document.write( "B = 9139 is the number of ways to pick the Republicans\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "A*B = 35990*9139 = <font size=4 color=red>328,912,610</font> is the number of ways to pick the entire committee. <BR>\n" );
document.write( "Order does not matter. <BR>\n" );
document.write( "This number is very close to 329 million. <BR>\n" );
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