Mathematics: 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?

There are 100 students at a school and three dormitories

Ответ:

kolibeilfuss

05.02.2022

Mathematics

$N = 7.0 *10^{44} \ Ways$

$U = 2.85 *10^{26}\ ways$

Step-by-step 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

$N = \frac{n!}{A! B!C!}$

=> $N = \frac{100!}{25! 35! 40!}$

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

=> $N = \frac{100!}{25! 35! 40!}$

$N = \frac{9.332622* 10^{157}}{[1.551121* 10^{25}]* [1.0333148* 10^{40}] * [8.1591528*10^{47}]}$

$N = 7.0 *10^{44} \ Ways$

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

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

$U = ^{50}C_{35} * ^{50}C_{25} + 1$