document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204819>Question 1204819</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 5 -letter <BR>\n" );
document.write( "words'' are formed using the letters A, B, C, D, E, F, G, how many such words are possible for each of the following conditions:<BR>\n" );
document.write( "(a) No condition is imposed.<BR>\n" );
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document.write( "(b) No letter can be repeated in a word.<BR>\n" );
document.write( "Your answer is : \r<BR>\n" );
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document.write( "(c) Each word must begin with the letter A and letters can be repeated.<BR>\n" );
document.write( "Your answer is : \r<BR>\n" );
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document.write( "(d) The second letter must be a vowel and no letters can be repeated.<BR>\n" );
document.write( "Your answer is : \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841328 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=841328\">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>16807</font><BR>\n" );
document.write( "(b) <font color=red>2520</font><BR>\n" );
document.write( "(c) <font color=red>2401</font><BR>\n" );
document.write( "(d) <font color=red>720</font>\r<BR>\n" );
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document.write( "Explanation for part (a)\r<BR>\n" );
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document.write( "There are 7 letters in the set {A,B,C,D,E,F,G}\r<BR>\n" );
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document.write( "When no condition is imposed, we can repeat letters. <BR>\n" );
document.write( "There are 7 choices and 5 slots to fill. Giving 7^5 = <font color=red>16807</font> different words possible. <BR>\n" );
document.write( "Most of these \"words\" won't be found in the dictionary, but are simply strings of the letters mentioned. \r<BR>\n" );
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document.write( "Explanation for part (b)\r<BR>\n" );
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document.write( "Unlike the previous part, we cannot repeat letters here.<BR>\n" );
document.write( "There are 7 choices for the first slot, then 6 for the next slot, and so on.\r<BR>\n" );
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document.write( "We have 7*6*5*4*3 = <font color=red>2520</font> different permutations possible.\r<BR>\n" );
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document.write( "Alternatively, use the nPr formula with n = 7 and r = 5.\r<BR>\n" );
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document.write( "Explanation for part (c)\r<BR>\n" );
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document.write( "The first slot is locked to letter A.<BR>\n" );
document.write( "The remaining 4 letters can be anything of the set {A,B,C,D,E,F,G} where repeats are allowed. <BR>\n" );
document.write( "The logic is similar to part (a) we've done earlier.<BR>\n" );
document.write( "We have 7^4 = <font color=red>2401</font> different words that start with A when repeat letters are allowed.\r<BR>\n" );
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document.write( "Explanation for part (d)\r<BR>\n" );
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document.write( "Given set = {A,B,C,D,E,F,G}<BR>\n" );
document.write( "vowels from that set = {A, E}\r<BR>\n" );
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document.write( "There are 2 choices for the second slot (either A or E)<BR>\n" );
document.write( "There are 7-1 = 6 choices for the first slot, due to one of the vowels already taken.<BR>\n" );
document.write( "Then 6-1 = 5 choices for the third slot, 5-1 = 4 choices for the fourth slot, and 4-1 = 3 choices for the last slot.\r<BR>\n" );
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document.write( "6*2*5*4*3 = <font color=red>720</font> permutations where a vowel is the second letter.<BR>\n" );
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