
Room Assignments
Suppose 4 people were randomly assigned into 9 rooms (numbering 1 to 9). Each person can be asigned equally likely and each room can hold more than one person.
a) What is the probability that from rooms 1 to 4 there are exactly one person each?
b) What is the probability that there are exactly 3 people in room 8?

Taking the problem at face value, it is possible to assign all four to the same room, then there are $\displaystyle 9^4$ ways to do it.
If part a means the only rooms 1,2,3,&4 have a person in them then there are $\displaystyle 4 !$ ways to do that.
There are $\displaystyle 4 \choose 3$ to choose three to put in room eight. There are then 8 ways to assign the other person.