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 #839433 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=839433\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "If we could tell the \"A\"s apart, then there would be 5! = 5*4*3*2*1 = 120 different five-letter permutations. \r<BR>\n" );
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document.write( "But because we cannot tell the \"A\"s apart, we must divide by 2 which corrects the erroneous double-count.\r<BR>\n" );
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document.write( "We'll have 120/2 = 60 different five-letter words possible.\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "We have these four possible cases:<ol><li>The letter \"A\" is used twice</li><li>The letter \"A\" is used exactly once in the 1st slot.</li><li>The letter \"A\" is used exactly once in the 2nd slot.</li><li>The letter \"A\" is not chosen at all.</li></ol>Case 1 has exactly one outcome. That outcome is AA.\r<BR>\n" );
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document.write( "Case 2 has 3 choices for the 2nd slot, which means we have the 3 permutations AT, AR, AI\r<BR>\n" );
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document.write( "Case 3 also has 3 outcomes. The logic is similar to case 2.\r<BR>\n" );
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document.write( "Case 4 has 3*2 = 6 permutations of the letters {T,R,I}\r<BR>\n" );
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document.write( "Add up the results<BR>\n" );
document.write( "1+3+3+6 = 13<BR>\n" );
document.write( "There are 13 different two letter sequences\r<BR>\n" );
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document.write( "The entire list is<ol><li>AA</li><li>AT</li><li>AR</li><li>AI</li><li>TA</li><li>TR</li><li>TI</li><li>RA</li><li>RT</li><li>RI</li><li>IA</li><li>IT</li><li>IR</li></ol>A combinatorics calculator I recommend is this<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html\">https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html</a><BR>\n" );
document.write( "It computes the nPr value, and also lists the different permutations. <BR>\n" );
document.write( "Refer to the \"List Them\" section on that page.\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "The previous part calculated 13 different two-letter words.\r<BR>\n" );
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document.write( "Let's calculate how many three-letter sequences are possible.\r<BR>\n" );
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document.write( "We have these cases<ol><li>The letter \"A\" is used twice.</li><li>The letter \"A\" is used exactly once.</li><li>The letter \"A\" is not chosen at all.</li></ol>\r<BR>\n" );
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document.write( "Case 1<BR>\n" );
document.write( "There are 3 slots to pick from where the letters T, R, or I will go.<BR>\n" );
document.write( "After that slot is chosen, the other two slots are locked in as \"A\"<BR>\n" );
document.write( "Therefore, we have 3*3 = 9 ways to do case 1.\r<BR>\n" );
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document.write( "Case 2<BR>\n" );
document.write( "There are 3 slots to pick for the letter \"A\".<BR>\n" );
document.write( "We have two more slots and 3 more letters to pick from to fill said slots. We'll have 3*2 = 6 permutations of those other letters.<BR>\n" );
document.write( "Overall we have 3*6 = 18 ways to carry out case 2.\r<BR>\n" );
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document.write( "Case 3<BR>\n" );
document.write( "There are 3*2*1 = 6 permutations of the letters {T,R,I}\r<BR>\n" );
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document.write( "Add up the results<BR>\n" );
document.write( "9+18+6 = 33<BR>\n" );
document.write( "We find there are 33 different three-letter words.\r<BR>\n" );
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document.write( "I'll let you do the scratch work to determine how many four-letter words are possible. <BR>\n" );
document.write( "You should find there are 60 different four-letter words possible.\r<BR>\n" );
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document.write( "To summarize we have...<ul><li>60 ways to form a five-letter word.</li><li>60 ways to form a four-letter word.</li><li>33 ways to form a three-letter word.</li><li>13 ways to form a two-letter word.</li></ul>There are 60+60+33+13 = 166 different words possible when we can only select letters from the list {A,T,A,R,I} where \"A\" can be used at most twice, but the other letters at most once.\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "(a) <font color=red>60</font><BR>\n" );
document.write( "(b) <font color=red>13</font><BR>\n" );
document.write( "(c) <font color=red>166</font><BR>\n" );
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