The question is short but it may be a little bit tricky:
Suppose we have a deck that contains n cards with n different rank (let's say ranks are 1, 2, ... , n), and then we randomly shuffle the deck, what's the probability that all cards' orders are different from their ranks?
Such as:
$1^{st}$ card cannot be the card with rank 1, and
$2^{nd}$ card cannot be the card with rank 2, and
... and
$n^{th}$ card cannot be the card with rank n.
I have done some simulations for a specific n (n=7), feel free to use the sample probability distribution to test against your thoughts:
> n <- rep(-1, 1000000)
> for (i in 1:length(n)) {
+ deck <- sample(1:7)
+ n[i] <- sum(sample(1:7) == 1:7)
+ }
> table(n) / length(n)
n
0 1 2 3 4 5 7
0.367319 0.368180 0.183299 0.062795 0.014050 0.004150 0.000207
