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 #752436 by greenestamps(13200)\"\" \"About 
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\n" ); document.write( "The statement of the problem is sufficiently unclear that different interpretations are possible.

\n" ); document.write( "If the 8 teachers are individually identifiable, then each of the 8 teachers has 4 possibilities of which school to be assigned to; the \"number of divisions\" is then 4^8.

\n" ); document.write( "A more common (and more interesting) problem is with the 8 teachers NOT individually identifiable, and the \"number of divisions\" is the number of different ways the 4 schools can get certain numbers of teachers -- for example, 3 teachers to school A, none to school B, 1 to school C, and 4 to school D.

\n" ); document.write( "So this kind of problem is one in which, if we let A, B, C, and D represent the numbers of teachers assigned to each school, we are looking for the number of solutions in whole numbers of the equation

\n" ); document.write( "\"A%2BB%2BC%2BD+=+8\"

\n" ); document.write( "This kind of equation comes up in a large number of similar types of problems. A well-known method for solving them is \"stars and bars\".

\n" ); document.write( "With the stars and bars method, you start with 8 stars, representing the 8 teachers to be divided among the schools:
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document.write( "    * * * * * * * *

\n" ); document.write( "To divide the teachers among the four schools, you add THREE bars as separator symbols; for example
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document.write( "    * * | * * | * * | * *    (2 teachers to each of the 4 schools)\r\n" );
document.write( "    * * * | | * | * * * *    (3; 0; 1; and 4}\r\n" );
document.write( "    * | * * * | * | * * *    (1; 3; 1; and 3)

\n" ); document.write( "Each different placement of the 3 separator symbols -- i.e., each different arrangement of the 8 stars and 3 bars -- represents one of the possible ways to divide the 8 teachers among the 4 schools.

\n" ); document.write( "By a well known counting principle, that number of different arrangements is

\n" ); document.write( "\"%2811%21%29%2F%28%288%21%29%283%21%29%29+=+165\"

\n" ); document.write( "...a vastly different answer with this interpretation of the problem...! \n" ); document.write( "
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