document.write( "Question 1165084: Susie has three identical apples, three identical oranges and three identical
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Algebra.Com's Answer #789793 by ikleyn(52781)\"\" \"About 
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Susie has three identical apples, three identical oranges and three identical
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document.write( "We should consider SEVERAL different cases.\r\n" );
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document.write( "Case 1).  There are 1 apple, 2 oranges and 3 pears among the selected 6 pieces of fruit\r\n" );
document.write( "          Short signature is (1a, 2o, 3p).\r\n" );
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document.write( "      +------------------------------------------------------------------------------------------------+\r\n" );
document.write( "      |   and 5 other \"isomorphic\" cases, making all possible 6 (six) permutations of this signature    |\r\n" );
document.write( "      +------------------------------------------------------------------------------------------------+\r\n" );
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document.write( "Case 2).  There are 3 apples, 1 orange and 2 pears among the selected 6 pieces of fruit.\r\n" );
document.write( "          Short signature is (3a, 1o, 2p).\r\n" );
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document.write( "Case 3).  There are 2 apples, 3 oranges and 1 pear among the selected 6 pieces of fruit.\r\n" );
document.write( "          Short signature is (2a, 3o, 1p).\r\n" );
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document.write( "Case 4).  There are 3 apples, 2 oranges and 1 pear among the selected 6 pieces of fruit.\r\n" );
document.write( "          Short signature is (3a, 2o, 1p).\r\n" );
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document.write( "Case 5).  There are 1 apple, 3 oranges and 2 pears among the selected 6 pieces of fruit.\r\n" );
document.write( "          Short signature is (1a, 3o, 2p)\r\n" );
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document.write( "Case 6).  There are 2 apples, 1 orange and 3 pears among the selected 6 pieces of fruit.\r\n" );
document.write( "          Short signature is (2a, 1o, 3p).\r\n" );
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document.write( "Case 7).  There are 2 apples, 2 oranges and 2 pears among the selected 6 pieces of fruit.\r\n" );
document.write( "          Short signature is (2a, 2o, 2p).\r\n" );
document.write( "          For this signature, all other permutations are identical and, therefore, are not considered.\r\n" );
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document.write( "Case 8).  There are 3 apples and 3 oranges.\r\n" );
document.write( "          Short signature is (3a, 3o, 0p).\r\n" );
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document.write( "      +--------------------------------------------------------------------------------------------------+\r\n" );
document.write( "      |   and 2 other \"isomorphic\" cases, making all possible 3 (three) permutations of this signature    |\r\n" );
document.write( "      +--------------------------------------------------------------------------------------------------+\r\n" );
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document.write( "Case 9).  There are 3 apples and 3 pears.\r\n" );
document.write( "          Short signature is (3a, 0o, 3p).\r\n" );
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document.write( "Case 10). There are 3 oranges and 3 pears.\r\n" );
document.write( "          Short signature is (0a, 3o, 3p).\r\n" );
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document.write( "It is clear that\r\n" );
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document.write( "    a)  all these 10 cases produce DIFFERENT sequences of 6 fruits, ordered in the line;\r\n" );
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document.write( "    b)  \"isomorphic\" cases 1) - 6)  produce EQUAL number of different 6-piece sequences\r\n" );
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document.write( "              so, calculating these 6 cases, it is enough to calculate the number of sequences for any one single signature\r\n" );
document.write( "              and then multiply it by 6 in the total sum;\r\n" );
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document.write( "    c)  \"isomorphic\" cases 8) - 10)  produce EQUAL number of different 6-piece sequences\r\n" );
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document.write( "              so, calculating these 3 cases, it is enough to calculate the number of sequences for any one single signature\r\n" );
document.write( "              and then multiply it by 3 in the total sum.\r\n" );
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document.write( "Now, the number of all different linear sequences for case 1) is  \"6%21%2F%281%21%2A2%21%2A3%21%29\" = \"720%2F%282%2A6%29\" = 60.\r\n" );
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document.write( "     I use the WELL KNOWN formula for arrangements of 6 items with 2 and 3 undistinguishable items.\r\n" );
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document.write( "Next, the number of all different linear sequences for case 7) is  \"6%21%2F%282%21%2A2%21%2A2%21%29\" = \"720%2F%282%2A2%2A2%29\" = 90.\r\n" );
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document.write( "     I use the SIMILAR formula for arrangements of 6 items with 2, 2 and 2 undistinguishable items.\r\n" );
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document.write( "Finally, the number of all different linear sequences for case 8) is  \"6%21%2F%283%21%2A3%21%29\" = \"720%2F%286%2A6%29\" = 20.\r\n" );
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document.write( "After that, taking into account everything that was said above, the final answer is\r\n" );
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document.write( "    N = 6*60 + 90 + 3*20 = 510.\r\n" );
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document.write( "ANSWER.  The total number of all different linear sequences (arrangements) in this problem is 510.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Dear tutor @Seutip (!)\r
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\n" ); document.write( "\n" ); document.write( "If I sometime come to fix your solutions or to make my notices, please do not consider it as my\r
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\n" ); document.write( "\n" ); document.write( "There is nothing personal in it - my admiration only (!)\r
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\n" ); document.write( "\n" ); document.write( "I am very glad that so prominent tutor came to this forum (!)\r
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