Permutation
A permutation is a selection where the order matters. Even if you pick the same three people, placing them in first, second, or third counts as a different case. So the first spot has n choices, the next one fewer at (n-1), and so on; multiply r of them and you get nPr. Drag the top sliders for n and r and the choices shrink at each spot, and nPr appears; use the round-table slider and circular permutations, where rotations count as the same, appear.
Picking the same people still yields more cases once you assign seats. A permutation is the count that keeps those seat orders, and the symbol nPr is the product of the leftovers as you fill r seats from n items, front to back. The first seat has n choices, the next n − 1, and so on through seat r; that product is the same as n! / (n − r)!. At a round table, turning the whole seating looks the same, so n rotations count as one and the value is (n − 1)!. Lining everyone up is the case r = n, which is n!. If some items match, you divide the full factorial by the factorials of those repeats. 0! = 1 because these formulas treat an empty product as 1.
The upper figure lays out n balls with numbers; only the first r are filled with color. Below, r slots run from the 1st onward, and each slot shows the leftover count n, n − 1, and so on. Dashed lines join the balls to the slots, and the bottom line writes nPr as that product. n runs from 2 to 8 and r from 1 to n. The starting values are n = 5 and r = 3. On the round-table figure, as many colored balls as the chosen count sit on a circle, with the word for a fixed seat beside the top place. The formula under the circle shows the value of (n − 1)!. The table count runs from 3 to 7 and starts at 4.
If you count n people around a table as if they were a line of n!, you double-count seatings that are just one rotation of each other. To treat those rotations as one seating you fix one person and arrange the rest, and that value is (n − 1)!. On a lining-up problem, a symbol that drops order throws away swapped front-and-back lines and leaves the count too small. Two people who must sit together are first treated as one block, then the block’s own front and back go into the product. In a listing with repeated items, skipping the denominator counts indistinguishable seats as different cases.