Twelve students sit around a circular table. Let three of the students be #A#, #B# and #C#. Find the probability that #A# does not sit next to either #B# or #C#?
Let's say that there are 12 seats and number them 1 - 12.
Let's put A into seat 2. This means B and C can't sit in seats 1 or 3. But they can sit everywhere else.
Let's work with B first. There are 3 seats where B can't sit and so therefore B can sit in one of the remaining 9 seats.
For C, there are now 8 seats where C can sit (the three that are disallowed by sitting on or near A and the seat occupied by B).
The remaining 9 people can sit in any of the remaining 9 seats. We can express this as
Putting it all together, we have:
But we want the probability that B and C don't sit next to A. We'll have A stay in the same seat - seat number 2 - and have the remaining 11 people arrange themselves around A. This means there are
Therefore, the probability that neither B nor C sit next to A is: