document.write( "Question 1199154: Consider all of the ways of ordering the 5 letters U,S,A,M,O. Suppose these are listed in a dictionary in alphabetical order, starting with AMOSU. What position In the list would USAMO be? \r
\n" ); document.write( "\n" ); document.write( "A. 97 B. 103 C. 109 D. 115 E. 120\r
\n" ); document.write( "\n" ); document.write( "(If your answer is not one of these answer choices, you should double check your work, as it is wrong.) \n" ); document.write( "

Algebra.Com's Answer #832855 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Consider all of the ways of ordering the 5 letters U,S,A,M,O.
\n" ); document.write( "Suppose these are listed in a dictionary in alphabetical order, starting with AMOSU.
\n" ); document.write( "What position In the list would USAMO be?
\n" ); document.write( "A. 97 B. 103 C. 109 D. 115 E. 120
\n" ); document.write( "(If your answer is not one of these answer choices, you should double check your work, as it is wrong.)
\n" ); document.write( "~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Consider the list of all alphabetically ordered words of the 5 letters U,S,A,M,O.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Before than U will appear in the 1st positions, all the words with U in the \r\n" );
document.write( "5th position, 4th position, 3rd position and 2nd position must appear.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The number of words with U in the 5th position is 4! = 24.\r\n" );
document.write( "The same is the number of words with U in the 4th position;\r\n" );
document.write( "                                              3rd position and\r\n" );
document.write( "                                              2nd positions.\r\n" );
document.write( "\r\n" );
document.write( "It gives 4*4! = 4*24 = 96 such preceding words.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next, with U in the 1st position, before than S will appear in the 2nd position,\r\n" );
document.write( "all the words, starting with U and with S in the 5th position, 4th position and \r\n" );
document.write( "3rd position must appear. The number of such words is 3*3! = 3*6 = 18.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "As soon as we have the word starting with \"US\" in the list, the next word will be USAMO.\r\n" );
document.write( "Its sequential number in the list is 96 + 18 + 1 = 115.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  The position of the word USAMO in the list is 115.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved, with full explanations.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "USAMO is the abbteviation for \"The United States of America Mathematical Olympiad\".\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The United States of America Mathematical Olympiad (USAMO) is a highly selective
\n" ); document.write( "high school mathematics competition held annually in the United States.
\n" ); document.write( "Since its debut in 1972, it has served as the final round of the American Mathematics Competitions.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "See this Wikipedia article \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "https://en.wikipedia.org/wiki/United_States_of_America_Mathematical_Olympiad#:~:text=The%20United%20States%20of%20America,of%20the%20American%20Mathematics%20Competitions.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );