document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199271>Question 1199271</A>: <I><SPAN STYLE=\"word-break: break-all;\"> All of the arrangements of the letters in the word ALGEBRA are written on a list, one of these arrangments selected at random. What is the probability that the arrangement contains a double-A </SPAN>\n" );
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document.write( "My interpretation of \"all of the arrangements of the letters in the word ALGEBRA\" is that all 7 letters are used in each arrangement.<br><BR>\n" );
document.write( "Here are two ways to find the probability that the two A's are next to each other when all 7 letters are used.<br><BR>\n" );
document.write( "(1) Counting numbers of permutations....<br><BR>\n" );
document.write( "The number of ways of arranging the 7 letters is 7!<br><BR>\n" );
document.write( "For the number of ways of having the two A's together, treat those two letters as a unit.  We are now arranging 6 items; there are 6! arrangements, and the two A's can be in either of two orders.  So the number of arrangements of the 7 letters with the two A's together is 2*6!.<br><BR>\n" );
document.write( "The probability of having the two A's together is then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2A6%21%29%2F7%21=2%2F7 ALIGN=MIDDLE  ALT=\"%282%2A6%21%29%2F7%21=2%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(2) Considering the different places in the string where the two A's are together....<br><BR>\n" );
document.write( "If the first A is in either the first or last position (probability 2/7), then there is only one of the other 6 positions where the second A can be (probability 1/6).  The probability for this case is (2/7)(1/6) = 2/42.<br><BR>\n" );
document.write( "If the first A is in any of the other 5 positions (probability 5/7), then the second A can be in either of two of the other 6 positions (either side of the first A -- probability 2/6).  The probability for this case is (5/7)(2/6) = 10/42.<br><BR>\n" );
document.write( "The overall probability is then 2/42 + 10/42 = 12/42 = 2/7.<br><BR>\n" );
document.write( "ANSWER: 2/7<br><BR>\n" );
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