Probability of person who sit in a circle

[desmond iking]

Homework Statement

there are 7 seats around a round table, and is labelled form A to G. Find the number of ways a committee of 3 teachers and 4 parents can sit around the table?

(i ) there is no restriction
my ans is 7!=5040

(ii) all teachers must sit together
the ans is 1008. can someone explain part ii please? my wokring is (6!/(3! x 4!) ) x 4! = 120...

Homework Equations

The Attempt at a Solution

 

[Dick]

Homework Statement

there are 7 seats around a round table, and is labelled form A to G. Find the number of ways a committee of 3 teachers and 4 parents can sit around the table?

(i ) there is no restriction
my ans is 7!=5040

(ii) all teachers must sit together
the ans is 1008. can someone explain part ii please? my wokring is (6!/(3! x 4!) ) x 4! = 120...

Homework Equations

The Attempt at a Solution

What's your logic for that working? First count the number of ways to pick a block of 4 adjacent chairs, then multiply by the number of ways you can seat the 4 teachers there and the three parents in the remaining 3 seats.

 

[desmond iking]

sorry my working should be 3! x 5! = 720 which means i arrange the teacher in 3 ! ways first , then i arrange the whole set of teacher into another 5! ways.

 

[haruspex]

sorry my working should be 3! x 5! = 720 which means i arrange the teacher in 3 ! ways first , then i arrange the whole set of teacher into another 5! ways.

Guessing here, but I think what you mean is that having fixed the order of the 3 teachers you treat them as a single entity. So now you have four parents and a block of teachers.
That doesn't quite work. Say you put the teacher block 'at' A. Is that at ABC, or GAB or FGA?
Having got the 3! for the block of teachers, think about how to arrange the parents as a block, then how to match the two blocks to the 7 chairs.