# Probability - Cominations and Integer Valued Vectors

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,...,xr) satisfying x1 + x2 + ... + xr = 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.

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Your'e making it too complicated

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

HallsofIvy