Final answer:
To find the number of ways to divide 8 new teachers among 4 schools, we use the stars and bars method, resulting in 165 possible divisions using the combinatorial formula C(11, 3).
Step-by-step explanation:
The question is asking to find out the number of ways 8 new teachers can be divided among 4 schools. This is a problem of combinatorics, a field of mathematics. To solve this, we can use the stars and bars combinatorial method, which is useful for dividing identical items (in this case, teachers) into distinct groups (schools).
Imagine the 8 teachers are stars (*) and we need 3 bars (|) to create 4 distinct groups. An example of one distribution could be **|***|**|*, which represents 2 teachers in the first school, 3 in the second, 2 in the third, and 1 in the fourth. The number of ways to arrange these stars and bars is the number of ways to choose 3 spots from 11 possible spots (8 stars plus 3 bars).
The formula for this calculation is the combination formula: C(n + k - 1, k), where n is the number of items to distribute and k is the number of groups minus 1. In this case, C(11, 3) or C(11, 8), as choosing where to place the bars is equivalent to choosing where to place the stars.
Thus, the calculation is C(11, 3) = 11! / (3! * (11 - 3)!) = 165 possible divisions. Therefore, there are 165 possible divisions of 8 new teachers among 4 schools.