document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203955>Question 1203955</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 56 Democratic senators and 44 Republican senators?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839903 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=839903\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "There are 56*55*54 = 166,320 permutations of selecting 3 Democrats from a pool of 56. <BR>\n" );
document.write( "We start at 56 and count down until we have filled 3 slots. Multiply the values along the way. \r<BR>\n" );
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document.write( "On a committee, no member outranks another. There aren't special named seats such as \"president\", \"VP\", \"treasurer\", etc.<BR>\n" );
document.write( "The lack of named seats indicates that order doesn't matter.<BR>\n" );
document.write( "A committee like {Alice, Bob, Carol} is the same as {Carol, Alice, Bob}.<BR>\n" );
document.write( "Because order doesn't matter on a committee, we divide by 3! = 3*2*1 = 6. <BR>\n" );
document.write( "There are 6 ways to rearrange {a,b,c}. We must divide by 6 to correct for the erroneous overcount.\r<BR>\n" );
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document.write( "(166,320)/6 = 27,720<BR>\n" );
document.write( "There are 27,720 different combinations of 3 Democrats from a pool of 56.\r<BR>\n" );
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document.write( "Another way to arrive at the value 27,720 is to use the nCr combination formula with n = 56 and r = 3<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "56 C 3 = (56!)/(3!*(56-3)!)<BR>\n" );
document.write( "56 C 3 = (56!)/(3!*53!)<BR>\n" );
document.write( "56 C 3 = (56*55*54*53!)/(3!*53!)<BR>\n" );
document.write( "56 C 3 = (56*55*54)/(3!)<BR>\n" );
document.write( "56 C 3 = (56*55*54)/(3*2*1)<BR>\n" );
document.write( "56 C 3 = 166320/6<BR>\n" );
document.write( "56 C 3 = 27720<BR>\n" );
document.write( "Pay careful attention to the fact, on the third to last step, we have \"56*54*53\" in the numerator and \"3*2*1\" in the denominator.\r<BR>\n" );
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document.write( "Through similar calculations we have 44C3 = 13,244 different trios of Republicans where we have select from a pool of 44.\r<BR>\n" );
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document.write( "Overall there are 27720*13244 = 367,123,680 different committees where order doesn't matter.\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>367,123,680</font><BR>\n" );
document.write( "(approximately 367 million)<BR>\n" );
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