document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206342>Question 1206342</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (a) How many different ways can 2 letters from the word SELECT be arranged?<BR>\n" );
document.write( "(b) How many different ways can 3 letters from the word SELECT be arranged? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843739 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=843739\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answers:<BR>\n" );
document.write( "(a) <font color=red size=4>21</font><BR>\n" );
document.write( "(b) <font color=red size=4>72</font>\r<BR>\n" );
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document.write( "Explanation for part (a)\r<BR>\n" );
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document.write( "Break things up into three cases:<BR>\n" );
document.write( "(1) Exactly zero 'E's are chosen<BR>\n" );
document.write( "(2) Exactly one 'E' is chosen<BR>\n" );
document.write( "(3) Exactly two 'E's are chosen\r<BR>\n" );
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document.write( "Case (1)<BR>\n" );
document.write( "The letters to pick from are {S,L,C,T}<BR>\n" );
document.write( "There are 4*3 = 12 different two letter words possible where \"E\" isn't chosen.<BR>\n" );
document.write( "Alternatively, you can use the nPr permutation formula with n = 4 and r = 2.<BR>\n" );
document.write( "Of course when I say \"word\", I mean it in quotes because much of these two-letter strings aren't words found in the dictionary.<BR>\n" );
document.write( "An example word of case (1) would be SL.\r<BR>\n" );
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document.write( "Case (2)<BR>\n" );
document.write( "Let's say \"E\" is in the first slot. There would be 4 words we can form which are: ES, EL, EC, ET<BR>\n" );
document.write( "We will also have 4 words with \"E\" in the second slot.<BR>\n" );
document.write( "That's 4+4 = 8 different words that have exactly one \"E\".<BR>\n" );
document.write( "An example word of case (2) would be SE.\r<BR>\n" );
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document.write( "Case (3)<BR>\n" );
document.write( "This is a trivial case of just one possibility. The word EE.\r<BR>\n" );
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document.write( "Add up the results:<BR>\n" );
document.write( "12+8+1 = <font color=red>21</font> is the final answer to part (a)\r<BR>\n" );
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document.write( "Here is the list of all 21 unique entries (7 rows, 3 columns)<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >SE</TD><TD >SL</TD><TD >SC</TD></tr><tr><TD >2</TD><TD >ST</TD><TD >ES</TD><TD >EL</TD></tr><tr><TD >3</TD><TD >EE</TD><TD >EC</TD><TD >ET</TD></tr><tr><TD >4</TD><TD >LS</TD><TD >LE</TD><TD >LC</TD></tr><tr><TD >5</TD><TD >LT</TD><TD >CS</TD><TD >CE</TD></tr><tr><TD >6</TD><TD >CL</TD><TD >CT</TD><TD >TS</TD></tr><tr><TD >7</TD><TD >TE</TD><TD >TL</TD><TD >TC</TD></tr>\n" );
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document.write( "The list was generated using this combinatorics calculator<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>\r<BR>\n" );
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document.write( "Explanation for part (b)\r<BR>\n" );
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document.write( "We'll have three cases:<BR>\n" );
document.write( "Case (1): Exactly 0 'E's are chosen<BR>\n" );
document.write( "Case (2): Exactly 1 'E' is chosen<BR>\n" );
document.write( "Case (3): Exactly 2 'E's are chosen<BR>\n" );
document.write( "We stop here since we cannot select 3 \"E\"s.\r<BR>\n" );
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document.write( "Case (1)<BR>\n" );
document.write( "There are 4 items to pick from in the set {S,L,C,T}<BR>\n" );
document.write( "That gives 4*3*2 = 24 ways to form a three-letter word without any \"E\"s in it.<BR>\n" );
document.write( "An example word of case (1) would be SLC.\r<BR>\n" );
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document.write( "Case (2)<BR>\n" );
document.write( "We have 3 places to put the \"E\".<BR>\n" );
document.write( "Then we have 4*3 = 12 ways to pick the other letters where order matters.<BR>\n" );
document.write( "3*12 = 36 ways to form a three-letter word with exactly one \"E\" in it.<BR>\n" );
document.write( "An example word of case (2) would be SEL.\r<BR>\n" );
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document.write( "Case (3)<BR>\n" );
document.write( "There are 3 slots to choose from for one of these letters {S,L,C,T}<BR>\n" );
document.write( "3*4 = 12 ways to form a three-letter word with exactly two \"E\"s in it.<BR>\n" );
document.write( "An example word of case (3) would be SEE.\r<BR>\n" );
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document.write( "Add up the results:<BR>\n" );
document.write( "24+36+12 = <font color=red>72</font> is the final answer to part (b)\r<BR>\n" );
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document.write( "Below is the list of all 72 unique entries.<BR>\n" );
document.write( "The list was generated with the previously mentioned link.<BR>\n" );
document.write( "The table has 9 rows and 8 columns.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >1</TD><TD >2</TD><TD >3</TD><TD >4</TD><TD >5</TD><TD >6</TD><TD >7</TD><TD >8</TD></tr><tr><TD >1</TD><TD >SEL</TD><TD >SEE</TD><TD >SEC</TD><TD >SET</TD><TD >SLE</TD><TD >SLC</TD><TD >SLT</TD><TD >SCE</TD></tr><tr><TD >2</TD><TD >SCL</TD><TD >SCT</TD><TD >STE</TD><TD >STL</TD><TD >STC</TD><TD >ESL</TD><TD >ESE</TD><TD >ESC</TD></tr><tr><TD >3</TD><TD >EST</TD><TD >ELS</TD><TD >ELE</TD><TD >ELC</TD><TD >ELT</TD><TD >EES</TD><TD >EEL</TD><TD >EEC</TD></tr><tr><TD >4</TD><TD >EET</TD><TD >ECS</TD><TD >ECL</TD><TD >ECE</TD><TD >ECT</TD><TD >ETS</TD><TD >ETL</TD><TD >ETE</TD></tr><tr><TD >5</TD><TD >ETC</TD><TD >LSE</TD><TD >LSC</TD><TD >LST</TD><TD >LES</TD><TD >LEE</TD><TD >LEC</TD><TD >LET</TD></tr><tr><TD >6</TD><TD >LCS</TD><TD >LCE</TD><TD >LCT</TD><TD >LTS</TD><TD >LTE</TD><TD >LTC</TD><TD >CSE</TD><TD >CSL</TD></tr><tr><TD >7</TD><TD >CST</TD><TD >CES</TD><TD >CEL</TD><TD >CEE</TD><TD >CET</TD><TD >CLS</TD><TD >CLE</TD><TD >CLT</TD></tr><tr><TD >8</TD><TD >CTS</TD><TD >CTE</TD><TD >CTL</TD><TD >TSE</TD><TD >TSL</TD><TD >TSC</TD><TD >TES</TD><TD >TEL</TD></tr><tr><TD >9</TD><TD >TEE</TD><TD >TEC</TD><TD >TLS</TD><TD >TLE</TD><TD >TLC</TD><TD >TCS</TD><TD >TCE</TD><TD >TCL</TD></tr>\n" );
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