document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203679>Question 1203679</A>: <I><SPAN STYLE=\"word-break: break-all;\"> contest consists of finding all of the code words that can be formed from the letters in the name \"ATARI.\" Assume that the letter A can be used twice, but the others at most once.<BR>\n" );
document.write( "A) How many five-letter words can be formed?<BR>\n" );
document.write( "B) How many two-letter words can be formed?<BR>\n" );
document.write( "C) How many words can be formed? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839454 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=839454\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "1-letter words....<BR>\n" );
document.write( "There are 4 different letters to choose from.<BR>\n" );
document.write( "The number of 1-letter words is 4.<br><BR>\n" );
document.write( "2-letter words....<BR>\n" );
document.write( "If both letters A are used, there is only 1 word -- AA<BR>\n" );
document.write( "If the letter A is not repeated, then there are 4 choices for the first letter and then 3 choices for the second, so there are 4*3=12 2-letter words without both letters A.<BR>\n" );
document.write( "The number of 2-letter words is 1+12 =13.<br><BR>\n" );
document.write( "3-letter words....<BR>\n" );
document.write( "If both letters A are used, then the combination of letters can be AAI, AAR, or AAT.  There are (3!)/(2!) = 3 arrangements of each of those, for a total of 9.<BR>\n" );
document.write( "If both letters A are not used, then there are 4 choices for the first letter, 3 for the second, and 2 for the third, for a total of 4*3*2 = 24.<BR>\n" );
document.write( "The number of 3-letter words is 9+24 = 33.<br><BR>\n" );
document.write( "4-letter words....<BR>\n" );
document.write( "If both letters A are used, then the combination of letters can be AAIR, AAIT, or AART.  There are (4!)/(2!) = 12 arrangements of each of those, for a total of 36.<BR>\n" );
document.write( "If both letters A are not used, then the letters are ATRI in any order; there are 4! = 24 of those.<BR>\n" );
document.write( "The number of 4-letter words is 36+24 = 60.<br><BR>\n" );
document.write( "5-letter words....<BR>\n" );
document.write( "The letters are ATARI in any order, the number of arrangements is (5!)/(2!) = 60.<BR>\n" );
document.write( "There are 60 5-letter words.<br><BR>\n" );
document.write( "Summary:<BR>\n" );
document.write( "1-letter words: 4<BR>\n" );
document.write( "2-letter words: 13<BR>\n" );
document.write( "3-letter words: 33<BR>\n" );
document.write( "4-letter words: 60<BR>\n" );
document.write( "5-letter words: 60<br><BR>\n" );
document.write( "Total number of words: 4+13+33+60+60 = 170<br><BR>\n" );
document.write( "ANSWERS:<BR>\n" );
document.write( "A) 5-letter words: 60<BR>\n" );
document.write( "B) 2-letter words: 13<BR>\n" );
document.write( "C) total number of words: 170<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );