Original Question:
Five people check identical suitcases. At the baggage claim, each person takes one of the five suitcases at random. What is the probability that every person ends up with the wrong suitcase?
Answer: 11/30
Solution:
Let’s start from one person, one suitcase. The probability is 0.
Then let n be the number of people, which is also the number of suitcases
If n = 2, P = 1⁄2
If n = 3, P = 1⁄3
If n = 4, P = 3⁄8
If n = 5, P = 11⁄30
If n = 6, P = 53⁄144
It is easy to observe that Pn = Pn-1 + (-1)n/n!
But what if n = ∞?
P = 1⁄2 - 1⁄6 + 1⁄24 - 1⁄120 + 1⁄720 - ...
Recall the Taylor’s Expansion:
ex = 1 + x + x2/2 + x3/6 + x4/24 + x5/120 + x6/720 + ...
When we plug in x = -1, we get that limn → +∞P = 1⁄e, and therefore the answer to infinity people with infinity suitcases is 1⁄e.