A nursing home randomly assigns residents to live in 5 homes

A nursing home randomly assigns residents to live in 5 homes. A) If 3 residents apply to live in these homes, what is the probability they are all assigned to the same home? B) if 3 residents apply to live in these homes, what is the probability they are all assigned to different home? C) If 15 residents apply to live in these homes, what is the probability that exactly 3 are assigned to each home? Assume that all homes are large and the same size. Please show all of your work so I can understand how to do the problem. Thanks

Solution

A) If 3 residents apply to live in these homes, what is the probability they are all assigned to the same home?

Answer :

3C1 * (1/5)^3
3 * 1/125
3/125

B) if 3 residents apply to live in these homes, what is the probability they are all assigned to different home?

Answer :

(5*4*3) / 5^3 ---> 60/125 ---> 12/25

C) If 15 residents apply to live in these homes, what is the probability that exactly 3 are assigned to each home?

Answer :

This is same as saying that take the first five and put them in each of the 5 homes...
15C5 * 5*4*3*2*1 ---> 360360 ways

Next five and put them in each of the 5 homes....
10C5 * 5! ---> 30240 ways

Final five and put them in each of the 5 homes....
5! ---> 120 ways

(360360 + 30240 + 120) / Total

(360360 + 30240 + 120) / 5^15

0.0000128 ---> third answer