Algebra.Com's Answer #830758 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
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document.write( "How many ways are there to distribute 15  distinct items into 5 distinct boxes with:
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document.write( "i) At least two empty box.
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document.write( "ii) No empty box.
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document.write( " In this my post, I consider case (ii), ONLY.\r
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document.write( " The \" stars and bars \" method works for undistinguishable balls and distinguishable boxes, ONLY.\r
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document.write( " It is clear from the logic of this method. Many (if not all) autoritative sources point it directly. \r
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document.write( " See, for example, this Wikipedia article
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document.write( " https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)\r
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document.write( " Therefore, it is inadequate and hopeless to apply the \" stars and bars \" method for the given problem,
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document.write( " where the balls are considered as DISTINGUISHABLE.\r
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document.write( " For it, another technique should be used. See my solution below.\r
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document.write( "The formula for the number of distributions of n distinguishable items in m distinguishable boxes is \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 spreadshhets 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( " Below are calculations for n= 15, m= 2 (fifteen balls in two boxes).\r\n" );
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document.write( " k (-1)^k combin(2,k) (2-k)^15 Separate addends\r\n" );
document.write( " of formula (1)\r\n" );
document.write( " 0 1 1 32768 32768\r\n" );
document.write( " 1 -1 2 1 -2\r\n" );
document.write( " 32766 <<<---=== Final sum F(15,2)\r\n" );
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document.write( " It is not so easy to check it manually that F(15,2) = 32766 is the correct number of different distributions\r\n" );
document.write( " of 15 distinguishable balls in 2 distinguishable boxes, but at least it seems likelihood.\r\n" );
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document.write( " And finally, below are calculations for n= 15, m= 5 (fifteen balls in five boxes, the requested case).\r\n" );
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document.write( " k (-1)^k combin(5,k) (5-k)^15 Separate addends\r\n" );
document.write( " of formula (1)\r\n" );
document.write( " 0 1 1 30517578125 30517578125\r\n" );
document.write( " 1 -1 5 1073741824 -5368709120\r\n" );
document.write( " 2 1 10 14348907 143489070\r\n" );
document.write( " 3 -1 10 32768 -327680\r\n" );
document.write( " 4 1 5 1 5\r\n" );
document.write( " 25292030400 <<<---=== Final sum F(15,5)\r\n" );
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document.write( "ANSWER. The number of all different distributions of 15 distinguishable balls in 5 distinguishable boxes is 25292030400.\r\n" );
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document.write( "Solved.\r
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document.write( "Although the formula is given in the cited references, I think, there are not so many people who know
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document.write( "how to approach to the problem, who know about existing of this formula, who know on how to solve this problem in a right way.\r
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document.write( "This problem is of much higher level than average high school and even of much higher level than
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document.write( "an ordinary Math circle at a school or at a local University level.\r
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document.write( "Only professional mathematicians working in Combinatorial Math know about existing this formula.\r
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document.write( "Another circle of persons who can be competent in the area, are those who are professionally trained
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document.write( "to participate in International Math Olympiads, as well as their trainers and their teachers.\r
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document.write( "Third circle of such people are those who know well the relevant popular Math literature.
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document.write( "I myself do belong to this last category.\r
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document.write( "////////////////\r
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document.write( "Here one more tutor came (@rippletable) and brought an incorrect solution.\r
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document.write( "Ignore it, since it is incorrect.\r
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