There are 100 students at a school and three dormitories, A, B, and C, with capacities 25, 35 an

step 1 Hello everyone. The question is that a class has 20 students, sorry, 20 students, 12 are female,

There are 100 students at a school and three dormitories, A,...

There are 100 students at a school and three dormitories, $\mathrm{A}, \mathrm{B}$, and $\mathrm{C}$, with capacities 25, 35 and 40 , respectively. (a) How many ways are there to fill the dormitories? (b) Suppose that, of the 100 students, 50 are men and 50 are women and that A is an all-men's dorm, $B$ is an all-women's dorm, and $C$ is co-ed. How many ways are there to fill the dormitories?

There are 100 students at a school and three dormitories, $\mathrm{A}, \mathrm{B}$, and $\mathrm{C}$, with capacities 25, 35 and 40 , respectively.
(a) How many ways are there to fill the dormitories?
(b) Suppose that, of the 100 students, 50 are men and 50 are women and that A is an all-men's dorm, $B$ is an all-women's dorm, and $C$ is co-ed. How many ways are there to fill the dormitories?

Key Concepts

Multinomial Coefficients

A generalization of binomial coefficients, multinomial coefficients are used when partitioning a set into more than two groups. They calculate the number of ways to divide a set into several groups of specified sizes, such as dividing 100 students among three dorms with capacities of 25, 35, and 40. This concept encapsulates the sequential application of multiple binomial selections.

Restricted Counting (with Additional Constraints)

Restricted counting deals with additional limitations on the choices available when partitioning or arranging sets. For example, when dorms have gender restrictions (all-men, all-women, or mixed), the count must account for these conditions by separately considering the available subgroups (men and women) and then combining the selections. This ensures that the assignments satisfy all specified criteria.

Combinatorial Selection

This concept involves selecting a subset from a larger set where the order of selection does not matter. In problems like filling dormitories, one needs to count the number of ways to choose groups (such as 25 out of 100 students) by using combination formulas. This forms the basis of calculating possibilities when assigning people to groups without considering the order within the groups.

Binomial Coefficients

Binomial coefficients, represented as C(n, k) or 'n choose k', count the number of ways to select k elements out of n without regard to order. They are essential in problems where a set is partitioned into different groups, such as choosing a fixed number of students to occupy one dormitory, and then proceeding similarly with the remaining students.

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