Permutations

[F19]

Any help is appreciated!

1) Nine students are to be seated at a circular table with fifteen chairs. If 4 students insist on sitting together but Rafael refuses to be seated by the group of friends, how many seating arrangements are possible?

Thanks!

 

[soroban]

Hello, F19!

A challenging problem . . . I think I got it . . .

1) Nine students are to be seated at a circular table with fifteen chairs.
If 4 students insist on sitting together but Rafael refuses to be seated by the group of friends,
how many seating arrangements are possible?

Let the four friends be \(\displaystyle A,B,C,D.\)
Duct-tape together: \(\displaystyle \boxed{ABCD}\)


Let Rafael sit in any chair ... it doesn't matter.
Call that chair #1, and the number the chairs clockwise around the table.

The group of four friends must not occupy chair #2 or chair #15.
They occupy #3-to-6, #4-to-7, #4-to-8, . . . , #11-to-14.
They have 9 choices.

And the four friends can be shuffled in: \(\displaystyle 4! = 24\) ways.

The other 4 students can occupy any of the remaining 10 chair.
\(\displaystyle \text{There are: }\:_{10}P_4 \:=\:5,040\text{ ways.}\)


\(\displaystyle \text{Therefore, there are: }\:9 \times 24 \times 5,040 \;=\;1,088,640\text{ seating arrangements.}\)