document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195051>Question 1195051</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In how many ways can a committee of three men and four women be formed from a group of 10 men and 10 women? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #827391 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=827391\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let's find out how many ways there are to pick the three men. <BR>\n" );
document.write( "n = 10 men<BR>\n" );
document.write( "r = 3 selections<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "10 C 3 = (10!)/(3!*(10-3)!)<BR>\n" );
document.write( "10 C 3 = (10!)/(3!*7!)<BR>\n" );
document.write( "10 C 3 = (10*9*8*7!)/(3!*7!)<BR>\n" );
document.write( "10 C 3 = (10*9*8)/(3!)<BR>\n" );
document.write( "10 C 3 = (10*9*8)/(3*2*1)<BR>\n" );
document.write( "10 C 3 = (720)/(6)<BR>\n" );
document.write( "10 C 3 = 120\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are 120 ways to pick the three men from a pool of ten men. Order does not matter. This is why we use the nCr combination formula and not the nPr permutation formula.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Order doesn't matter on a committee like this because no member outranks another. No members hold special titles such as \"president\", \"VP\", \"secretary\", etc. Each seat is the same.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Follow similar steps for the women.<BR>\n" );
document.write( "n = 10 women<BR>\n" );
document.write( "r = 4 selections<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "10 C 4 = (10!)/(4!*(10-4)!)<BR>\n" );
document.write( "10 C 4 = (10!)/(4!*6!)<BR>\n" );
document.write( "10 C 4 = (10*9*8*7*6!)/(4!*6!)<BR>\n" );
document.write( "10 C 4 = (10*9*8*7)/(4!)<BR>\n" );
document.write( "10 C 4 = (10*9*8*7)/(4*3*2*1)<BR>\n" );
document.write( "10 C 4 = (5040)/(24)<BR>\n" );
document.write( "10 C 4 = 210\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are 210 ways to pick the four women from a pool of ten women. Like before, order doesn't matter.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Side note: The values 120 and 210 can be found in Pascal's Triangle. Look at the row that starts with 1,10,...<BR>\n" );
document.write( "The 10C3 = 120 is the fourth item in that row, while 10C4 = 210 is the fifth item. The general item 10Cr is item number r+1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Anyways, we found there are 120 ways to pick the men and 210 ways to pick the women.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Overall there are 120*210 = <u><font color=red>25200 ways</font></u> to form the seven person committee.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );