Coaching Centre
Arrangements in a circle
Count arrangements around a circle by fixing one person, arrange groups around a circle, and find probabilities for circular arrangements.
Fix one person
Around a round table, only the positions of people relative to each other matter. Rotating everyone one seat to the left gives the same arrangement. So each arrangement in a line of $n$ seats corresponds to $n$ rotations of one circular arrangement.
In how many ways can $6$ people sit around a round table?
Groups around a circle
Seven people sit at a round table. In how many ways can they sit if Ali and Bea sit together? If they sit apart?
Four men and four women sit at a round table, alternating. How many ways?
Probability around a circle
Seven people sit at random at a round table. Find the probability that Ali and Bea sit together.
In how many ways can $6$ people sit around a round table?
$5!$.
Seven people at a round table. How many ways if Ali and Bea sit together?
$5! \times 2$.
How many ways if they sit apart?
$720 - 240$.
Four men and four women sit alternately at a round table. How many ways?
$3! \times 4!$.
Seven people sit at random at a round table. Find the probability that Ali and Bea sit together.
$\dfrac{2}{6}$.
Why is the count for $n$ people in a circle $(n - 1)!$ and not $n!$?
Each circular arrangement has $n$ rotations.
Six different beads are threaded on a necklace. How many different necklaces?
$\dfrac{5!}{2}$.
In how many ways can $2$ people sit at a round table with $2$ seats?
$1! = 1$: each sits opposite the other.
Five people sit at a round table. The probability that a particular person is next to Ali is:
Seat Ali. Of the $4$ other seats, $2$ are next to Ali.
In how many ways can $8$ people sit around a round table?
$7!$.
New numbers every time. Keep going until you get five in a row.