MHB How Many Ways to Fit Six People in Four Rooms?

[anemone]

Here is this week's POTW:

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In how many different ways can six people be accommodated in the four rooms, if each room can accommodate a maximum of four people?

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[anemone]

Congratulations to kaliprasad for his correct solution, which you can find as follows::)

Spoiler

Six persons can be accommodated in 4 rooms in $4^6=4096$ ways without any constraints.

We need to rule out the number of ways 6 or 5 persons can be accommodated in a room.

Number of ways 6 persons can be accommodated in one room = 4 as in 4 ways we can choose the room for all persons.

If one room has 5 persons, and the other one room has one person. Then the 5 persons can be chosen in 6 ways, and the room for 5 persons can be chosen in 4 ways and then room for one person can be chosen in 3 ways. So the number of ways for such cases $= 6 \times 4 \times 3 = 72$.

Therefore the number of ways atmost 4 persons can be in a room = 4096- 4 - 72 = 4020.