How many ways can 12 people be seated about a round table?

If there are specific places around the table, there will be 12!

If not, then place 11!
(Suppose we look at one particular arrangement, everybody get up and move one to the right. The arrangement has not changed. But we could do this 12 times.
That is way we have only 11!, since 12!/12 = 11!)

To determine the number of ways that 12 people can be seated around a round table, we can use the concept of permutations. Since the table is round, the relative positions of the people are important.

The number of ways to arrange 12 people in a row is given by the formula for permutations, which is n! (n factorial). Here, n is the number of people, so n! = 12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1.

However, when the people are seated around a round table, we need to account for the fact that the arrangement can be rotated and still be considered the same. In other words, the starting point on the table doesn't matter.

To account for this, we divide the total number of arrangements by the number of ways the table can be rotated. Since there are 12 people, there are 12 possible starting points. So, the number of ways to arrange 12 people around a round table is:

12! / 12 = 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 39,916,800

Therefore, there are 39,916,800 different ways to seat 12 people around a round table.