96.4k views
0 votes
There are 80 indistinguishable students in the freshman class of Indy Integirls Academy who need to be distributed into 3 distinct classrooms such that each classroom has at most 30 students. In how many ways can the classes be formed?

User Ladan
by
8.2k points

1 Answer

7 votes

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.

User Darrelltw
by
8.9k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories