133496
How many ways are there to arrange 9 people in a line such that no one is standing in their correct position?
Relevant Theorem: In combinatorics, the Derangement Formula, also known as the subfactorial or !n, is used to count the number of derangements (or permutations) of a set of n elements where no element appears in its original position. In other words, it calculates the number of ways to rearrange a set such that none of the elements are in their initial positions.

The derangement formula can be defined recursively as follows:

!0 = 1
!1 = 0
!n = (n-1)(!(n-1) + !(n-2)) for n > 1

Alternatively, it can be expressed using the inclusion-exclusion principle:

!n = n! (1/0! - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)

where n! denotes the factorial of n, which is the product of all positive integers up to n.

For example, let's find the number of derangements for a set of 3 elements {A, B, C}:

!3 = 3! (1/0! - 1/1! + 1/2!) = 6 (1 - 1 + 1/2) = 6 * 1/2 = 3

There are 3 derangements for this set: {B, C, A}, {C, A, B}, and {A, C, B}.