Probability - Cominations and Integer Valued Vectors

[AsianMan]

This problem comes from Sheldon Ross's book "A First Course in Probability (6th ed)."

There are 5 hotels in a certain town. If 3 people check into hotels in a day, what is the probability that they each check into a different hotel?


Attempt at a solution:

There are 5C3 = 10 different combinations of hotels where each individual person picks a different hotel.

I also decided that there were 7C4 = 35 possible ways for 3 individuals to choose from the 5 hotels, if more than 1 can stay in the same hotel. I got this answer because there are (n+r-1)C(r-1) distinct nonnegative integer-valued vectors (x₁,x₂,...,x_r) satisfying x₁ + x₂ + ... + x_r = n, where n = 3 and r = 5.

Therefore, I got 10/35 as my answer, but the answer is actually .48 (rounded?)

Interestingly, I got very close this answer mistakenly at first by dividing 5C3 by 7C2.

 

[Random Variable]

Your'e making it too complicated

$$\frac {P(5,3)}{5^{3}}$$

 

[HallsofIvy]

The first person arrives and checks into any hotel. The second person arrives and checks into a hotel. What is the probability that person checks into a different hotel? The third person arrives. What is the probability this person checks into yet a different hotel? The probability that they check into three different hotels is the product of those two probabilities.. This is exactly the same as Random Variable gives- although, Random Variable, it would be better not to just "give" answers. Especially in the "coursework and homework sections".

 

[AsianMan]

Wow, I'm embarrassed.

Thanks guys!

BTW, is there any way to do it the way that I was doing it?