A fancy bed and breakfast inn has 5 rooms, each with a distinctive color-coded decor. One day 5 friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than 2 friends per room. In how many ways can the innkeeper assign the guests to the rooms?
A. 2100
B. 2220
C. 3000
D. 3120
E. 3125

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qwlavender qwlavender · Answer 1 · 2022-11-28T16:32:21+01:00

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 )

assign frequencies to rooms

= 5 × 4 × 3! ( 5 2 )

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:

https://brainly.com/question/2408784

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