document.write( "Question 1205233: In a cars carrier, how many ways are there to put 8 cars labeled as {A, B , ..., H} in 4 containers labeled as { 1, 2, 3, 4 }? Each container must include at least one car.
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Algebra.Com's Answer #841895 by ikleyn(52908) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "This problem (first part) asks in how many ways 8 distinguishable items can be distributed
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document.write( "                    Solution\r\n" );
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document.write( "The formula for the number of distributions of n distinguishable items in m distinguishable boxes \r\n" );
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document.write( "        F(n,m) = .    (1)\r\n" );
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document.write( "The sources for this formula are these references  \r\n" );
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document.write( "    Feller - An Introduction to Probability Theory and its Applications, Vol I, 3ed, 1968,\r\n" );
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document.write( "    Chen Chuan-Chong, Koh Khee-Meng - Principles and Techniques in Combinatorics, 1992,\r\n" );
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document.write( "    Anderson - A first course in combinatorial Mathematics, 2001.\r\n" );
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document.write( "To make calculations using this formula, I prepared Excel spreadsheets for some different values n and m.\r\n" );
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document.write( "    Below are calculations for n= 3, m= 2  (three balls in two boxes).\r\n" );
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document.write( "	k	(-1)^k	combin(2,k)	(2-k)^3	   Separate addends\r\n" );
document.write( "                                                   of formula (1)\r\n" );
document.write( "	0	  1	   1	           8	      8\r\n" );
document.write( "	1	 -1	   2	           1	     -2\r\n" );
document.write( "					              6    <<<---=== Final sum F(3,2)\r\n" );
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document.write( "    You can check it manually that F(3,2) = 6 is the correct number of different distributions\r\n" );
document.write( "    of 3 distinguishable balls in 2 distinguishable boxes.\r\n" );
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document.write( "    Below are calculations for n= 4, m= 2  (four balls in two boxes).\r\n" );
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document.write( "        k	(-1)^k	combin(2,k)	(2-k)^4	   Separate addends\r\n" );
document.write( "                                                   of formula (1)\r\n" );
document.write( "        0	  1	   1	          16	     16\r\n" );
document.write( "        1	 -1	   2	           1	     -2\r\n" );
document.write( "				                     14    <<<---=== Final sum F(4,2)\r\n" );
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document.write( "    You can check it manually that F(4,2) = 14 is the correct number of different distributions\r\n" );
document.write( "    of 4 distinguishable balls in 2 distinguishable boxes.\r\n" );
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document.write( "    And finally, below are calculations for n= 8, m= 4  (8 balls in 4 boxes, the requested case).\r\n" );
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document.write( "        k	(-1)^k	combin(4,k)	(4-k)^8	   Separate addends\r\n" );
document.write( "                                                   of formula (1)\r\n" );
document.write( "        0	  1	   1	         65536	     65536\r\n" );
document.write( "        1	 -1	   4	          6561	    -26244\r\n" );
document.write( "        2	  1	   6	           256	      1536\r\n" );
document.write( "        3	 -1	   4	             1	        -4\r\n" );
document.write( "				                     40824    <<<---=== Final sum F(8,4)\r\n" );
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document.write( "ANSWER.  The number of all different distributions of 8 distinguishable objects in 4 different boxes is 40824.\r\n" );
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