AMC 10 Daily Practice Round 1

There are ways to randomly assign volunteers to different Olympic venues.

To ensure that each venue has at least one volunteer, we consider two possible cases:

Case : One venue gets volunteers, while the other two venues each get volunteer.

Select which venue gets volunteers: .

Select which volunteers go to that venue:

Assign the remaining volunteers to the remaining venues:

Total ways for this case: .

Case : One venue gets volunteer, while the other two venues each get volunteers.

Select which venue gets volunteer:

Select which volunteer goes to that venue:

Select out of the remaining volunteers for the second venue:

The last volunteers automatically go to the remaining venue.

Total ways for this case:

.

The total number of favorable assignments is: .

Thus, the probability that each venue receives at least one volunteer is:

Choose .