document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1115915>Question 1115915</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In how many ways you can arrange the letters of the word THINK so that the T and the K are separated by at least one letter?\r<BR>\n" );
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document.write( "Thank you,, </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #730765 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13208)</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=730765\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "It useful mental exercise to consider different ways of counting the number of arrangements, seeing that the different ways give the same answer.  I immediately see three different ways to do the counting.<br><BR>\n" );
document.write( "(1) List the pairs of positions the T and K can be in; for each of those pairs, the remaining 3 letters can be arranged in 3!=6 different ways.<BR>\n" );
document.write( "1 and 3; 1 and 4; 1 and 5; 2 and 4; 2 and 5; and 3 and 5; and the same pairs in the opposite order.<br><BR>\n" );
document.write( "Answer: 12 pairs of positions for the T and K; 6 arrangements of INK for each, so 12*6 = 72 arrangements.<br><BR>\n" );
document.write( "(2) For each possible position of the letter T, determine the number of possible positions for letter K; again for each of those, the remaining letters can be arranged in 6 different ways.<BR>\n" );
document.write( "T first --> 3 possible positions for K<BR>\n" );
document.write( "T second --> 2 possible positions for K<BR>\n" );
document.write( "T third --> 2 possible positions for K<BR>\n" );
document.write( "T fourth --> 2 possible positions for K<BR>\n" );
document.write( "T fifth --> 3 possible positions for K<br><BR>\n" );
document.write( "That makes a total of 12 different pairs of positions for T and K.<br><BR>\n" );
document.write( "And again the answer is 12*6 = 72 arrangements.<br><BR>\n" );
document.write( "(3) Subtract the number of arrangements in which the T and K are NOT separated by at least one letter from the total number of arrangements of the 5 letters.<br><BR>\n" );
document.write( "The total number of arrangements is 5!=120.<br><BR>\n" );
document.write( "There are 4 pairs of positions in which the T and K can be together: 1 and 2, 2 and 3, 3 and 4, and 4 and 5.<BR>\n" );
document.write( "In each of those pairs, the T and K can be in either of two orders.<BR>\n" );
document.write( "And again the remaining three letters can be arranged in 6 different ways.<br><BR>\n" );
document.write( "Total number of arrangements in which T and K are together: 4*2*6 = 48<br><BR>\n" );
document.write( "Number of arrangements in which T and K are separated by at least one letter: 120-48 = 72. </SPAN>\n" );
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