Question

There are 100 students at a school and three dormitories A, B, and C with capacities of 25, 35, and 40, respectively.

Required:
a. How many ways are there to fill up the dormitories?
b. Suppose that, of the 100 students, 50 are men and 50 arewomen and that A is an all-men's dorm, B is an all-women's dorm andC is co-ed. How many ways are there to fill thedormitories?

Answer 1

a

`N = 7.0 *10^(44) \ Ways`

b

`U  = 2.85 *10^(26)\  ways`

Explanation:

From the question we are told that

The number of students are n = 100

The number of dormitories is k = 3

The capacity of the first dormitory is A = 25

The capacity of the second dormitory is B = 35

The capacity of the third dormitory is c = 40

Generally the number of way to fill the dormitory up is mathematically represented as

`N  =  (n!)/(A! B!C!)`

=>
`N  =  (100!)/(25! 35! 40!)`

Here ! stands for factorial, so we will be making use of the factorial functionality in our calculators to evaluated the above equation

=>
`N  =  (100!)/(25! 35! 40!)`

`N = (9.332622* 10^(157))/([1.551121* 10^(25)]* [1.0333148* 10^(40)] * [8.1591528*10^(47)])`

`N = 7.0 *10^(44) \ Ways`

From the question we are told that there are 50 men and 50 women and

A is all-men's dorm and B is all-women's dorm while C is co-ed

So

When A is filled , the number of men that will be remaining to fill dorm C is 50-25 = 25

While when B is filled the number of women that will be remaining to fill dorm C is 50-35 = 15

Generally the number of ways there to fill the dormitories is equivalent to the number of ways of selecting the 25 men and 35 women to fill dormitory A and B plus one more way which is filling dorm C with the remaining students this is mathematically represented as

`U  =  ^(50)C_(35) * ^(50)C_(25)  +  1`

Here C stands for combination hence we will be making use of the combination functionality in our calculators

`U  = 2.250829575* ^(12)  * 1.264106064 * 10^(14) +  1`

=>
`U  = 2.85 *10^(26)\  ways`