Question 1134889
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The statement of the problem is sufficiently unclear that different interpretations are possible.<br>
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.<br>
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.<br>
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<br>
{{{A+B+C+D = 8}}}<br>
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".<br>
With the stars and bars method, you start with 8 stars, representing the 8 teachers to be divided among the schools:<br><pre>
    * * * * * * * *</pre>
To divide the teachers among the four schools, you add THREE bars as separator symbols; for example<br><pre>
    * * | * * | * * | * *    (2 teachers to each of the 4 schools)
    * * * | | * | * * * *    (3; 0; 1; and 4}
    * | * * * | * | * * *    (1; 3; 1; and 3)</pre>
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.<br>
By a well known counting principle, that number of different arrangements is<br>
{{{(11!)/((8!)(3!)) = 165}}}<br>
...a vastly different answer with this interpretation of the problem...!