Question

4.7. How many ways are there to seat n couples around a circular table so that no couple sits together? Express your answer as using summation notation. (Note: Rotated seatings are considered to be the same, so that abcd is the same as dabc, but reflected seatings are considered to be different, so that abcd is not the same as adcb).

Answer 1

ix).(m,,s), (m".) Suppose that there are n couples are named like (im Now, place m,m,.,m around the table in one way. Now, there are places in between m,m,...m where can sit. Now, f can sit everywhere except 2 places around m So, , can sit in (n-2) ways s, can sit in (n-3) ways because one place is already taken by and f, can't it around m Same ways, S can sit in (n-4) ways. So on f2 can sit in 1 way and so f and f,. So, the number of ways to sit around the table such that no couple sit together is n-2 n-3)....2)(1)(1)(1)-n-2)! ways.