【Example 6】In the senior year, 10 classes are holding graduation exams, and 10 teachers are assigned to proctor, among whom 5 are class advisors and they cannot proctor their own classes. How many different ways are there to arrange the proctoring?
The current example is of the same type as the previous one, so by similar analysis, we can determine that there are
$$
C_{5}^{0} \cdot A_{10}^{10}-C_{5}^{1} \cdot A_{9}^{9}+C_{5}^{2} \cdot A_{8}^{8}-C_{5}^{3} \cdot A_{7}^{7}+C_{5}^{4} \cdot A_{6}^{6}-C_{5}^{5} \cdot A_{5}^{5}
$$

ways to arrange the invigilation that meet the requirements. Now let's discuss a more advanced problem. If Example 6 is changed to "10 class teachers are now assigned to invigilate, but none can invigilate their own class," the original problem-solving approach still applies. In this scenario, there would be
$$
C_{10}^{0} \cdot A_{10}^{10}-C_{10}^{1} \cdot A_{9}^{9}+C_{10}^{2} \cdot A_{8}^{8}-\cdots+C_{10}^{10} \cdot A_{1}^{1}
$$

ways to arrange the invigilation that meet the requirements. However, since the first two terms in the above result are equal, they can be canceled out, resulting in the number of allowed invigilation arrangements being
$$
10!\left(\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\cdots-\frac{1}{9!}+\frac{1}{10!}\right)
$$