In how many ways can 8 People be Seated around a Circular Table?

In how many ways can 8 People be Seated around a Circular Table?

To determine how many ways 8 people can be seated around a circular table, use the concept of circular permutations.

Circular Permutations:

In a circular arrangement, the position of the first person is fixed to account for the rotational symmetry, meaning that one specific arrangement can be rotated and still be considered the same arrangement.

Therefore, the number of distinct ways to arrange n people around a circular table is given by (n−1)!

Solution

For 8 people, the number of distinct arrangements is calculated as follows

(8−1)! = 7!

7! = 7×6×5×4×3×2×1 = 5040

The number of ways in which 8 people can be seated around a circular table is 5040.

Circular Permutations

Circular permutations refer to the arrangement of objects in a circular pattern. Unlike linear arrangements, circular arrangements have no fixed starting or ending point, making them unique in their own way.

Key Points:

• In circular permutations, the focus is on arranging objects in a loop or circle.
• The primary characteristic is that rotating the entire arrangement does not create a new permutation. For example, arranging A, B, and C around a circle is the same as arranging them in the order of B, C, A, or C, A, B.

Formula:

The formula to calculate the number of circular permutations of n distinct objects is (n−1)!

This is because one position is fixed to avoid counting rotations of the same arrangement as different permutations.

Applications:

• Circular permutations are often used in problems involving seating arrangements, such as around a round table.
• They are also used in arranging beads in a necklace, forming circular designs, and solving puzzles involving circular arrangements.

Understanding circular permutations is essential for various fields, such as mathematics and event planning. It helps in solving problems where the order of arrangement in a loop is significant, and it avoids over-counting equivalent arrangements due to rotations.

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