Respuesta :
In 2220 ways the innkeeper can assign the guests to the rooms i.e. Option B is correct.
As per the question,
- Number of rooms = 5
- Number of friends = 5
- The rooms are distinctive colour-coded decor i.e. the rooms have to be distinct.
There are Five friends and Five rooms,
⇒ 0 rooms of 2, 5 rooms of 1
⇒ 1 room of 2, 3 rooms of 1
⇒ 2 rooms of 2, 1 room of 1
Guests can be allotted rooms by the innkeeper as
1, 1, 1, 1, 1, 1, 2, 2, or 1, 1, 1, 2
- To assign ( 1, 2, 2 ) = ways to assign guests to the rooms × ways to assign frequencies to rooms
= 5 × ( 4 2 ) × ( 5 2 ) ( 3 2 )
- To assign ( 1, 2, 2 ) = ways to assign guests to the rooms × ways to
assign frequencies to rooms
= 5 × 4 × 3! ( 5 2 )
- To assign ( 1, 1, 1, 1, 1 ) = 5!
Total number of ways ( N ) = To assign ( 1, 2, 2 ) + To assign ( 1, 2, 2 ) +
To assign ( 1, 1, 1, 1, 1 )
= 5 × ( 4 2 ) × ( 5 2 ) ( 3 2 ) + 5 × 4 × 3! ( 5 2 ) +
5!
= 5 × ( 4 2 ) × ( 5 2 ) ( 3 2 ) + 120 ( 5 2 ) + 120
N = 2220 ways
Therefore, The innkeeper can assign guests to the rooms in 2220 ways with no more than two friends per room i.e. option B is valid.
To know more about combinations refer to:
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