Q.In how many different ways can 8 friends sit around a table?

A circular permutation is an arrangement of objects around a closed loop (a round table, a garland, a necklace) rather than in a straight row, and it is counted differently from a row-arrangement because a circular arrangement has no fixed starting point: rotating every object one seat around produces the SAME relative arrangement, since everyone still has exactly the same left- and right-neighbour as before. If $n$ distinct objects were instead arranged in a row, there would be $n!$ linear arrangements; but each single circular arrangement, when 'cut open' at any one of its $n$ positions, corresponds to exactly $n$ different linear arrangements (one for each place the cut could be made) that are all really the same circular seating. So the number of distinct circular arrangements of $n$ distinct objects is $\dfrac{n!}{n}=(n-1)!$ — equivalently, fix any one object's seat to remove the rotational duplicate-counting, and arrange the remaining $n-1$ objects relative to it in $(n-1)!$ ways. Two important refinements: if $m$ of the $n$ objects are identical, the count becomes $\dfrac{(n-1)!}{m!}$ (dividing out their internal rearrangements just as with row permutations). And if a clockwise seating and its exact mirror-image (anticlockwise) seating are considered to be the SAME arrangement — because, again, every person ends up with the same pair of neighbours either way — the count is further halved to $\dfrac{(n-1)!}{2}$. Typical problem-solving techniques layered on top of the basic formula include bundling objects that must sit together into one block before applying the circular formula to the resulting (smaller) set of units, and — for 'no two of a certain group are adjacent' conditions — seating the unrestricted group first to create gaps around the circle, then permuting the restricted group into a chosen subset of those gaps.