document.write( "Question 1134889: If 8 teachers are to be divided among 4 schools, how many divisions are possibles? \n" ); document.write( "

Algebra.Com's Answer #752414 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " This problem has FANTASTICALLY beautiful solution.\r
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document.write( "Consider the linear expression  X + Y + Z + W of four variables X, Y, Z and W, and consider its 8-th degree .\r\n" );
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document.write( "If you FOIL this expression and collect like terms, you will get the sum\r\n" );
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document.write( "       = .    (1)\r\n" );
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document.write( "consisting of monoms    with integer coefficients  C(i,j,k,m) that depend on the sets of non-negative integer indexes (i, j, k, m).\r\n" );
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document.write( "The sum  (1)  is spread (is distributed) over all such sets of non-negative integer indexes that i+j+k+m = 8.\r\n" );
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document.write( "We can interpret each monom     (or each set (i,j,k,n) )  as a distribution sample of 8 teachers (= 8 objects) \r\n" );
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document.write( "    i teachers in the school X;\r\n" );
document.write( "    j teachers in the school Y;\r\n" );
document.write( "    k teachers in the school Z  and\r\n" );
document.write( "    m teachers in the school W\r\n" );
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document.write( "among 4 school (=4 boxes).  Doing in this way,  we do not personalize the teachers - we distinct and consider only \"signatures\"  (i,j,k,n), \r\n" );
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document.write( "attaching and connecting them to school X, Y, Z and W.\r\n" );
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document.write( "But if we personalize the teachers, we will get the coefficients  C(i,j,k,m) showing HOW MANY personalized combinations of teachers \r\n" );
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document.write( "we will have for each given signature (i,j,k,m).\r\n" );
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document.write( "So, each given coefficient  C(i,j,k,m) shows how many personalized combinations are there for each given numerical signature (i,j,k,m).\r\n" );
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document.write( "But in this problem we are not interested in knowing each and every coefficient C(i,j,k,m) individually and separately - \r\n" );
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document.write( "we are interested only to know the sum of all coefficients C(i,j,k,m).\r\n" );
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document.write( "The last observation is that this sum is nothing else as the value of   at X= 1, Y= 1, Z=1 and W= 1.\r\n" );
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document.write( "Thus the sum of all coefficients  \r\n" );
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document.write( "     =  = .\r\n" );
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document.write( "And it gives the ANSWER to the problem's question:  \r\n" );
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document.write( "    There are    ways to distribute 8 teachers among 4 school.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved, completed, answered and explained.\r
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\n" ); document.write( "\n" ); document.write( "For more general problem\r
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document.write( "    In how many ways N distinguishable objects  can be distributed among n different boxes\r\n" );
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\n" ); document.write( "\n" ); document.write( "the solution and the answer are similar: in $\"n%5EN\"$ ways.\r
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document.write( "    This problem is, obviously, far above the average school level.\r\n" );
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document.write( "    It is typical problem of the Math Circle level at a respectful university.\r\n" );
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