A travel bureau estimates that when 20 tourists go to a resort with ten hotels. find the probability that no hotel is left vacant when the first group of 20 tourists arrives.

The solution for this problem is:

One way to solve is to using the "stars and bars" approach, but this could be the other one:

x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 = 20

then use (20+10-1) C (10-1), this will result to 29C9.

Then find the way to distribute x

as a result, (20-10+10-1) C (10-1) = 19C9

Thus, the probability is 19C9/29C9 or 0.0093