document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205549>Question 1205549</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A club consists of 16 men and 17 women. In how many ways can they choose a president, vice president, treasurer, and secretary, along with an advisory committee of six people? </SPAN>\n" );
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document.write( "16 men + 17 women = 33 people total\r<BR>\n" );
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document.write( "33*32*31*30 = 982080 ways to pick the president, VP, treasurer and secretary. Order matters because the seats or positions have different names.<BR>\n" );
document.write( "The basic idea is to start at 33 and count down by 1 until 4 slots are accounted for. We multiply each value along the way.<BR>\n" );
document.write( "Alternatively, you can use the nPr formula with n = 33 and r = 4.\r<BR>\n" );
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document.write( "After 4 people are chosen for those named seats there are 33-4 = 29 people left and there are 6 committee seats to fill.<BR>\n" );
document.write( "Unlike the previous 4 positions, order will not matter with these new 6 seats. <BR>\n" );
document.write( "This is because each person on the 6-person committee is of equal rank. <BR>\n" );
document.write( "Note how none of the 6 seats are named. \r<BR>\n" );
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document.write( "We have n = 29 people and r = 6 seats to fill.<BR>\n" );
document.write( "Using the nCr combination formula we get:<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "29 C 6 = (29!)/(6!*(29-6)!)<BR>\n" );
document.write( "29 C 6 = (29!)/(6!*23!)<BR>\n" );
document.write( "29 C 6 = (29*28*27*26*25*24*23!)/(6!*23!)<BR>\n" );
document.write( "29 C 6 = (29*28*27*26*25*24)/(6!)<BR>\n" );
document.write( "29 C 6 = (29*28*27*26*25*24)/(6*5*4*3*2*1)<BR>\n" );
document.write( "29 C 6 = 342014400/720<BR>\n" );
document.write( "29 C 6 = 475020\r<BR>\n" );
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document.write( "Pay close attention to the fact \"29*28*27*26*25*24\" is in the numerator of that nCr calculation above. The \"29*28*27*26*25*24\" calculates the permutation npr(29,6). We divide by 6! = 6*5*4*3*2*1 = 720 because this is the number of ways to rearrange any group of 6 people.\r<BR>\n" );
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document.write( "We found that...<ul><li>982080 ways to pick president, VP, treasurer and secretary (order matters)</li><li>475020 ways to pick the 6-person committee (order doesn't matter)</li></ul>There will be 982080*475020 = <font color=red>466,507,641,600</font> (roughly 466.5 billion) different ways to select all 10 people.\r<BR>\n" );
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document.write( "Effectively we just computed: npr(33,4)*ncr(29,6)\r<BR>\n" );
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document.write( "Alternatively you can compute it like this: ncr(33,6)*npr(27,4)<BR>\n" );
document.write( "Why does this work? Because we can form the committee of 6 people first, and then we'd have 33-6 = 27 people left to pick from to select the president, VP, treasurer, and secretary. \r<BR>\n" );
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document.write( "Answer: <font color=red size=4>466,507,641,600</font><BR>\n" );
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