document.write( "Question 1184895: If a permutation is chosen random from the letters “aaabbbccc” what is the probability that it begins with at least 2 a’s \n" ); document.write( "

Algebra.Com's Answer #815612 by ikleyn(52776) $\"\"$ $\"About$

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\n" ); document.write( "If a permutation is chosen random from the letters “aaabbbccc” what is the probability
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document.write( "(1)  We will consider distinguishable arrangements of 9 given letters.\r\n" );
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document.write( "     The total number of all such arrangements is   =  = 1680.\r\n" );
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document.write( "(2)  The number of all distinguishable arrangements starting with 3 a's is \r\n" );
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document.write( "          =  = 20.\r\n" );
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document.write( "(3)  The number of all distinguishable arrangements starting with  exactly  2 a's is equal (OBVIOUSLY) \r\n" );
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document.write( "     to the number of all distinguishable arrangements of the 7 (seven letter) \"abbbccc\", where \"a\" is not \r\n" );
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document.write( "     in the first (leftmost) position.\r\n" );
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document.write( "     The number of such arrangements is equal to the number of all distinguishable arrangements of these 7 letters\r\n" );
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document.write( "     (which is   = 140)    the number of all those distinguishable arrangements of these 7 letters,\r\n" );
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document.write( "     where \"a\" is in the first position. The latter number is   =  = 20.\r\n" );
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document.write( "     THUS,  the number of all distinguishable arrangements starting with  exactly  2  a's is equal to 140 - 20 = 120.\r\n" );
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document.write( "(4)  Finally, the number of all distinguishable arrangements starting with at least 2 a's is  20 + 120 = 140.\r\n" );
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document.write( "(5)  THEREFORE, the probability under the problem's question is  P =  =  =  = 0.08333... = 8.333...% .    ANSWER\r\n" );
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