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Seating Arrangements Calculator

Enter the number of people and seats to count seating arrangements.

Result

6,720

How it is calculated

The tool counts ordered selections, the permutations P(n, r) = n! ÷ (n − r)! = n × (n−1) × … × (n−r+1). Here n is the number of people available and r the number of seats in a row; who sits where matters, so swapping two people gives a new arrangement. It multiplies r descending factors rather than computing full factorials. Inputs are rounded to the nearest whole number, r cannot exceed n, and n is limited to 170; otherwise no result is shown. Results of 1015 or more switch to scientific notation (for example 2.4329e+18). Round tables, couples who must sit together and banned neighbours are not modelled.

Worked example

Eight guests are available and five chairs stand in a line, so n = 8 and r = 5. The first chair can be filled in 8 ways, the second in 7, the third in 6, the fourth in 5 and the fifth in 4. P(8, 5) = 8 × 7 × 6 × 5 × 4 = 6,720 different seating orders. If all eight guests had to be seated (r = 8), the count would jump to 8! = 40,320.

Try this example →

Common mistakes

  • Using the tool for a round table. Rotating everyone around a circle gives the same seating, so a circle of n people has (n−1)! arrangements, far fewer than this tool's n!.
  • Mixing it up with combinations. If you only want to choose which 5 of 8 people attend and the order does not matter, the count is C(8,5) = 56, not 6,720.
  • Entering more seats than people. With r greater than n the tool gives no result.
  • Expecting every real constraint to be honoured. Couples seated together, separated rivals and fixed head-table places are not included.
  • Typing a decimal such as 8.4 and expecting it to count differently; the tool rounds it to 8.

About this calculator

Seating r people chosen from a group of n in a row, where order matters, is counted by the permutation formula P(n,r) = n! / (n−r)! — the number of ways to fill the first seat times the ways to fill the second given the first is taken, and so on, for r seats. When r equals n, this reduces to n!, every arrangement of the whole group.

Teachers use this calculator to demonstrate permutations concretely in combinatorics lessons, event and wedding planners use it when working out how many distinct seating orders are possible for a head table, and students use it to check homework problems that ask for the number of ways to arrange people or objects in a line.

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