The probability of everyone getting a new table?

Question

We have 24 students in a classroom each with their own table. We decide to rearrange the seats and we give everyone a random place.

What is the chance that no one sits back at his former table?

Answer 1

It turns out that there s a recursive formula

Let S(n) be the number of switches with n people

S(1) = 0

S(n) = n S(n-1) + (-1)^n

Thus, S(24) = 2.28250211305339E+23

As 24! = 6.20448401733239E+23

P(no one sits at the same table) = 0.367879441171442

It should be noted that the probability approaches 1/e

See http://libai.math.ncu.edu.tw/webclass/statistics/probability/notes/ch3_sec5_p3/index.html for a derivation.

Answer 2

There are 24! ways to arrange the tables.

There is a 1/24! chance of getting the exact same arrangement.