Permutations and Combinations: Common Mistakes and Fixes (3)

Question

In how many ways can $7$ persons be seated around a round table such that two specific persons $A$ and $B$ never sit next to each other?

Solution — Step by Step

Why This Works

The complement principle is the standard PnC weapon: rather than counting “not adjacent” directly (which means choosing $A$‘s seat, then carefully picking $B$‘s non-neighbouring seats — easy to miscount), count the easier opposite event “adjacent” and subtract from total.

The “treat together as a unit” trick reduces an adjacency problem to a permutation of fewer items. The internal swap (here, $A$-$B$ vs $B$-$A$) is a separate factor.

Alternative Method

Direct counting: fix $A$‘s seat (since it’s circular, this kills the rotational symmetry — gives $6!$ for the rest, but we’re working with $(n-1)!$ already). Now $B$ has $6$ remaining seats, of which $2$ are adjacent to $A$. So $B$ has $4$ non-adjacent choices. The remaining $5$ persons can be arranged in $5!$ ways:

Same answer through different reasoning. Always cross-check using both methods on the day of the exam.

Common Mistake