There are seven performers who will present their comedy acts this weekend at a comedy club. How many different ways are there to schedule their appearances?

There are 5040 different ways to schedule their appearances.

Explanation:

Let A, B, C, D, E, F, G be the slots in which the given performs 1,2,3,4,5,6,7 can perform

For A we have 7 choices to select among (1,2,3,4,5,6,7)

For B we have 6 choices to select among (2,3,4,5,6,7)

For C we have 5 choices to select among (3,4,5,6,7)

For D we have 4 choices to select among (4,5,6,7)

For E we have 3 choices to select among (5,6,7)

For F we have 2 choices to select among (6,7)

For G we have 1 choice to select (7)

Thus total different choices are 7!=5040