document.write( "Question 1210250: The number of ways to arrange 4 green balls, 3 red balls, and 2 white balls in a straight line such that no two balls of the same color are adjacent is equal to:
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Algebra.Com's Answer #851753 by ikleyn(52777) $\"\"$ $\"About$

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\n" ); document.write( "The number of ways to arrange 4 green balls, 3 red balls, and 2 white balls in a straight line
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document.write( "Place 4 green balls along the straight line.\r\n" );
document.write( "Make gaps between these four green balls.\r\n" );
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document.write( "     You will have three gaps between the 4 green balls \r\n" );
document.write( "PLUS one gap on the left before these balls \r\n" );
document.write( "PLUS one gap on the right after these balls.\r\n" );
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document.write( "We will place 3 red balls and 2 white balls in these 3 + 2 = 5 gaps.\r\n" );
document.write( "We MUST place some of these 3 + 2 = 5 balls in the gaps between the green balls\r\n" );
document.write( "and we CAN place some of these 5 balls in the gaps before and/or after 4 green balls.\r\n" );
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document.write( "                It is the key idea to the problem' solution.\r\n" );
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document.write( "The rest is simply an implementation of this idea and detailed consideration/outlining/listing/counting \r\n" );
document.write( "of all possible cases.\r\n" );
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document.write( "    +----------------------------------------------------------------------------------------+\r\n" );
document.write( "    |           Let the gaps between the 4 green balls are numbered 1, 2, 3;                 |\r\n" );
document.write( "    |   the gap before green balls has number 0 and the gap after green balls has number 4.  |\r\n" );
document.write( "    +----------------------------------------------------------------------------------------+\r\n" );
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document.write( "(1)  This is the case when all 3 red balls and 2 white balls are placed in 3 gaps between the 4 green balls.\r\n" );
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document.write( "     We have these sub-cases\r\n" );
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document.write( "               gap  #       1    2    3\r\n" );
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document.write( "        balls in the gaps   RWR  R    W,  \r\n" );
document.write( "                            R    RWR  W,\r\n" );
document.write( "                            R    W    RWR,   \r\n" );
document.write( "        PLUS\r\n" );
document.write( "                            RWR  W    R,  \r\n" );
document.write( "                            W    RWR  R,\r\n" );
document.write( "                            W    R    RWR,   \r\n" );
document.write( "        PLUS\r\n" );
document.write( "                            WRW  R    R,  \r\n" );
document.write( "                            R    WRW  R,\r\n" );
document.write( "                            R    R    WRW.   \r\n" );
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document.write( "     It gives  3 + 3 + 3 = 9 possible arrangements.\r\n" );
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document.write( "     We also have these other sub-cases\r\n" );
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document.write( "               gap #        1    2    3\r\n" );
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document.write( "        balls in the gaps   2    2    1,   where  '2'  is  (R,W)  or (W,R) independently at every '2' appearance.\r\n" );
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document.write( "                            2    1    2,\r\n" );
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document.write( "                            1    2    2.   It gives  = 12 possible arrangements.\r\n" );
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document.write( "    So, case (1) gives 9 + 12 = 21 different possible arrangements.\r\n" );
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document.write( "(2)  This is the case when all 3 red and 2 white balls are placed in 5 gaps, one ball in each gap.\r\n" );
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document.write( "     It gives   =  = 10 different possible arrangements for case (2).\r\n" );
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document.write( "(3)  This is the case when 2 balls of different colors are placed in the gap '0'; \r\n" );
document.write( "     three remaining balls of 5 = 3R + 2W balls are placed in gaps 1, 2, and 3.\r\n" );
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document.write( "     It gives these arrangements\r\n" );
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document.write( "               gap  #       0   1    2    3   4\r\n" );
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document.write( "        balls in the gaps  RW   R    R    W,\r\n" );
document.write( "                           RW   R    W    R,\r\n" );
document.write( "                           RW   W    R    R,      (3 arrangements)\r\n" );
document.write( "        PLUS\r\n" );
document.write( "                           WR   R    R    W,\r\n" );
document.write( "                           WR   R    W    R,\r\n" );
document.write( "                           WR   W    R    R.      (3 arrangements).\r\n" );
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document.write( "    It gives 3 + 3 = 6  different possible arrangements for case (3).\r\n" );
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document.write( "(4)  This is the case when 2 balls of different colors are placed in the gap '4'; \r\n" );
document.write( "     three remaining balls of 5 balls (3R + 2W) are placed in gaps 1, 2, and 3.\r\n" );
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document.write( "     This case is SYMMETRIC to case (3).  It gives 6 other arrangements, symmetric to case (3).\r\n" );
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document.write( "(5)  This is the case when 1 ball R or W is placed in the gap '0'; \r\n" );
document.write( "     four remaining balls of 5 = 3R + 2W balls are placed in gaps 1, 2, and 3, \r\n" );
document.write( "     so as two balls (R and W) are placed into the same gap.\r\n" );
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document.write( "     It gives these arrangements\r\n" );
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document.write( "               gap  #       0   1    2    3   4\r\n" );
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document.write( "        balls in the gaps   R   R    W    RW,\r\n" );
document.write( "                            R   W    RW   R,\r\n" );
document.write( "                            R   RW   R    W,    \r\n" );
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document.write( "                            R   W    R    RW,\r\n" );
document.write( "                            R   R    RW   W,\r\n" );
document.write( "                            R   RW   W    R,    \r\n" );
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document.write( "                            R   R    W    WR,\r\n" );
document.write( "                            R   W    WR   R,\r\n" );
document.write( "                            R   WR   R    W,    \r\n" );
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document.write( "                            R   W    R    WR,\r\n" );
document.write( "                            R   R    WR   W,\r\n" );
document.write( "                            R   WR   W    R      (3*4 = 12 arrangements)\r\n" );
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document.write( "        PLUS\r\n" );
document.write( "                            W   R    R    RW,\r\n" );
document.write( "                            W   R    RW   R,\r\n" );
document.write( "                            W   RW   R    R,    \r\n" );
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document.write( "                            W   R    R    WR,\r\n" );
document.write( "                            W   R    WR   R,\r\n" );
document.write( "                            W   WR   R    R.     (3*2 = 6 arrangements)\r\n" );
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document.write( "    It gives 12 + 6 = 18  different possible arrangements for case (5).\r\n" );
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document.write( "(6)  This is the case when 1 ball R or W is placed in the gap '4'; \r\n" );
document.write( "     four remaining balls of 5 = 3R + 2W balls are placed in gaps 1, 2, and 3, \r\n" );
document.write( "     so as two balls (R and W) are placed into the same gap.\r\n" );
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document.write( "     This case is SYMMETRIC to case (5).  It gives 18 other arrangements, symmetric to case (5).\r\n" );
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document.write( "(7)  From cases (1), (2), (3), (4), (5) and (6),  we have, in all,  \r\n" );
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document.write( "         21 + 10 + 6 + 6 + 18 + 18 = 79 different possible arrangements.\r\n" );
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document.write( "ANSWER.  Doing this way, I counted 79 different possible arrangements.\r\n" );
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\n" ); document.write( "\n" ); document.write( "After completing my solution, I posted this problem to Google AI.\r
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\n" ); document.write( "\n" ); document.write( "Google AI tried to solve, but chose a wrong strategy from the very beginning (the inclusion-exclusion principle).\r
\n" ); document.write( "\n" ); document.write( "This strategy does not work for this problem (or requires much more delicate treatment),
\n" ); document.write( "so their solution immediately ran into insurmountable difficulties.\r
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\n" ); document.write( "\n" ); document.write( "Here is the link to this Google AI solution of 05/02/2025\r
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\n" ); document.write( "\n" ); document.write( "https://www.google.com/search?q=The+number+of+ways+to+arrange+4+green+balls%2C+3+red+balls%2C+and+2+white+balls+in+a+straight+line+such+that+no+two+balls+of+the+same+color+are+adjacent+is+equal+to%3A&rlz=1C1CHBF_enUS1071US1071&oq=The+number+of+ways+to+arrange+4+green+balls%2C+3+red+balls%2C+and+2+white+balls+in+a+straight+line++such+that+no+two+balls+of+the+same+color+are+adjacent+is+equal+to%3A&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIGCAEQRRg8MgYIAhBFGDzSAQkxNTg2ajBqMTWoAgiwAgHxBX64uq3p3LSl8QV-uLqt6dy0pQ&sourceid=chrome&ie=UTF-8\r
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\n" ); document.write( "\n" ); document.write( "On contrary, my methodology in this my post allows to break through the wall
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