Matt's four cousins are coming to visit. There are four identical rooms that they can stay in. If any number of the cousins can stay in one room, how many different ways are there to put the cousins in the rooms?
You can solve this using sticks and stones strategy.
However I am going to do case work because its simpler
Case 1: There is maximum 1 person in each room
Obviously 4! so 24 for case 1
Case 2: There are maximum 2 people in each rooms
So we can list possiblities
If ONLY 2 or 0 people can be in a room, then it is 2! => 4
If 2, 1, and 0 people can be in a room, then it is 3! => 6
So 10 cases for case 2
Case 3: There are Maximum 3 people in each room
Just like case 2 we can list possibilities
If ONLY 3, 1, and 0 people can be a room then it is 3! => 6
Thats all for case 3 so we have 6 because some of our possibilities were already done for case 2
Case 4: There are 4 people in one room with three other empty rooms
To find the answer we won't multiply the cases we will add them.
4+6+10+24 = 44 ways we can put 4 people in 4 bedrooms.