Answer: There are 6 ways to form the classes.
Step-by-step explanation: We need to distribute 80 indistinguishable students into 3 distinct classrooms such that each classroom has at most 30 students. Let the number of students in the three classrooms be x, y, and z, respectively. Then we have:
x + y + z = 80 (since the students are indistinguishable)
We want to find the number of non-negative integer solutions to this equation, subject to the condition that each variable is at most 30.
We can represent this problem using generating functions as follows:
The generating function for each variable is:
(1 + x + x^2 + ... + x^30) (since each variable can take on values from 0 to 30)
The generating function for the number of solutions is the product of the three generating functions:
(1 + x + x^2 + ... + x^30)^3
We need to find the coefficient of x^80 in this generating function, which will give us the number of ways to form the classes.
Using the binomial theorem, we can expand the generating function as follows:
(1 + x + x^2 + ... + x^30)^3 = (1 - x^31)^-3
Expanding the above using the binomial theorem, we get:
(1 - x^31)^-3 = ∑(n+2)C(2)x^(31n)
where ∑(n+2)C(2) represents the sum of the binomial coefficients (n+2)C(2) for n ranging from 0 to infinity.
To find the coefficient of x^80, we need to set n = 2, since 31n must be less than or equal to 80.
Thus, the coefficient of x^80 is (2+2)C(2) = 6.
Therefore, there are 6 ways to form the classes.