perms and combs qn (1 Viewer)

[Masaken]

Q: In how many ways can 6 students and 4 teachers be seated around a circular table if all teachers are separated?

Didn't get the question right originally (did 5! x 4!) originally, and drew a circle that fixed a student and then placed the teachers in places between the students.

Working out shows this, but I'm confused. Why did they draw 12 slots when there's only 10, and why 6P4?

 

[carrotsss]

The dashes are places where teachers could be, not actual seats. Correct me if I’m wrong but the working seems to indicate:

- Fix one student, lay the rest of the students around (5! ways the non-fixed students can be ordered)
- Place the teachers in the remaining empty spaces between students (6 places, need to pick 4 & order matters so 6P4)

So you get 6P4*5!

Hope this helps, although I must admit this isn’t how I would approach the question.

 

[Lith_30]

The only factor you forgot to take into account was that there were 6 spots that the teachers could sit in (as seen in the diagram), but only 4 teachers.

So you just have to multiply what you already had by to account for the additional arrangements. Which gives the same answer

 

[Deem_Skills]

The dashes are places where teachers could be, not actual seats. Correct me if I’m wrong but the working seems to indicate:

- Fix one student, lay the rest of the students around (5! ways the non-fixed students can be ordered)
- Place the teachers in the remaining empty spaces between students (6 places, need to pick 4 & order matters so 6P4)

So you get 6P4*5!

Hope this helps, although I must admit this isn’t how I would approach the question.

6P4 because first, there's 6 ways of seating the first teacher, then 5 ways of seating the second, 4 ways for third and 3 ways for the fourth which comes out as 6*5*4*3 = 6P4

 

[Masaken]

The only factor you forgot to take into account was that there were 6 spots that the teachers could sit in (as seen in the diagram), but only 4 teachers.

So you just have to multiply what you already had by to account for the additional arrangements. Which gives the same answer

that makes so much sense (and honestly probably how i would approach the question if it came up in an exam). thank you