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JEE Advanced 2020 · Paper 2, Mathematics question 14

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is ______

Answer and worked solution

Official answer: 1080

With every room holding 1 or 2 people, six people in four rooms must be split 2,2,1,12,2,1,1.

Rooms. Choose the two rooms that get two people: (42)=6\binom42=6 ways.

People. Assign the six people to the rooms:
6!2! 2! 1! 1!=180 ways.\frac{6!}{2!\,2!\,1!\,1!}=180\text{ ways}.

Total. 6×180=10806\times180=1080.

Answer: 1080

Topics:Distribution and arrangements
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