document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210159>Question 1210159</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the United States, there are 50 states.  Each state is represented by 2 senators.  In how many ways can we form a committee with 5 senators, in which no two of the senators are from the same state? </SPAN>\n" );
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document.write( "A similar question is found <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1169916.html\">here</a><BR>\n" );
document.write( "I recommend checking it out and using it as a tool to solve your current problem. If you're still stuck, then read on.\r<BR>\n" );
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document.write( "There are 50*2 = 100 senators total.<BR>\n" );
document.write( "For the first slot we have 100 choices.<BR>\n" );
document.write( "For the second slot we have 100-2 = 98 choices. <BR>\n" );
document.write( "We do not have 99 choices since we exclude the second senator from whatever state was chosen as the first slot.<BR>\n" );
document.write( "Otherwise, this committee could potentially have two senators from the same state.\r<BR>\n" );
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document.write( "We keep this process of subtracting 2 until we account for all five slots.<BR>\n" );
document.write( "The five slots have these choices: 100, 98, 96, 94, 92\r<BR>\n" );
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document.write( "If order mattered then we would have 100*98*96*94*92 = 8,136,038,400 different committees. <BR>\n" );
document.write( "Roughly 8.136 billion.\r<BR>\n" );
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document.write( "However, it appears that none of the seats have a title (such as \"chairman\" or \"treasurer\"), so this means a committee like ABCDE is the same as ACBDE. <BR>\n" );
document.write( "Order does <u>not</u> matter.\r<BR>\n" );
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document.write( "How many ways are there to scramble the letters in ABCDE? <BR>\n" );
document.write( "There are 5 letters and 5*4*3*2*1 = 120 permutations.<BR>\n" );
document.write( "So we must divide that 8.136 billion figure by 120 to arrive at the correct final answer.\r<BR>\n" );
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document.write( "8136038400/120 = <font color=red>67800320</font><BR>\n" );
document.write( "When using commas to make the number more readable, it would look like this 67,800,320. This is approximately 67.8 million. \r<BR>\n" );
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document.write( "A bit of random trivia: If you were to form a different committee every second, then it would take about 784.7 days (a little over 2 years) for you to try all 67.8 million options.\r<BR>\n" );
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document.write( "Answer: <font color=red>67800320</font><BR>\n" );
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