Six people are to sit at a circular table. Two of the people are not to sit immediately beside each other. Find the number of ways that the six people can be seated.

EITHER

with no restrictions six people can be seated in \(5! = 120\) ways     A1

we now count the number of ways in which the two restricted people will be sitting next to each other

call the two restricted people \({p_1}\) and \({p_2}\)

they sit next to each other in two ways     A1

the remaining people can then be seated in \(4!\) ways     A1

the six may be seated \({p_1}\) and \({p_2}\) next to each other) in \(2 \times 4! = 48\) ways     M1

\(\therefore \) with \({p_1}\) and ( \({p_2}\) not next to each other the number of ways \( = 120 - 48 = 72\)     A1     N3

[5 marks]

OR

person \({p_1}\) seated at table in \(1\) way    A1

\({p_2}\) then sits in any of \(3\) seats (not next to \({p_1}\) )     M1A1

the remaining \(4\) people can then be seated in \(4!\) ways     A1

\(\therefore \) number ways with \({p_1}\) not next to \({p_2} = 3 \times 4! = 72\) ways      A1     N3

Note: If candidate starts with \(6!\) instead of \(5!\), potentially leading to an answer of \(432\), do not penalise.

[5 marks]

Very few candidates provided evidence of a clear strategy for solving such a question. The problem which was set in a circular scenario was no more difficult than an analogous linear one.